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Composition sequences and semigroups ofMobius transformations
Matthew Jacques
MMath (Sheffield)
MSc (Open)
A thesis submitted to
School of Mathematics and Statistics
The Open University
for the degree of Doctor of Philosophy
January 23, 2017
Declaration
I confirm that the material contained in this thesis is the result of independent work,
except where explicitly stated, and with the exception of Chapters 2, 3 and 5, which are
the result of joint work with Ian Short. None of it has previously been submitted for a
degree or other qualification to this or any other university or institution.
Matthew Jacques
January 23, 2017
i
Abstract
Motivated by the theory of Kleinian groups and by the theory of continued fractions,
we study semigroups of Mobius transformations. Like Kleinian groups, semigroups have
limit sets, and indeed each semigroup is equipped with two limit sets. We find that limit
sets have an internal structure with features similar to the limit sets of Kleinian groups
and the Julia sets of iterates of analytic functions. We introduce the notion of a semidis-
crete semigroup, and find that this property is akin to the discreteness property for groups.
We study semigroups of Mobius transformations that fix the unit disc, and lay the foun-
dations of a theory for such semigroups. We consider the composition sequences generated
by such semigroups, and show that every such composition sequence converges pointwise
in the open unit disc to a constant function whenever the identity element does not lie in
the closure of the semigroup. We establish various results that have counterparts in the
theory of Fuchsian groups. For example we show that aside from a certain exceptional fam-
ily, any finitely-generated semigroup S is semidiscrete precisely when every two-generator
semigroup contained in S is semidiscrete. We show that the limit sets of a nonelementary
finitely-generated semidiscrete semigroup are equal (and non-trivial) precisely when the
semigroup is a group. We classify two-generator semidiscrete semigroups, and give the
basis for an algorithm that decides whether any two-generator semigroup is semidiscrete.
We go on to study finitely-generated semigroups of Mobius transformations that map the
unit disc strictly within itself. Every composition sequence generated by such a semigroup
converges pointwise in the open unit disc to a constant function. We give conditions that
determine whether this convergence is uniform on the closed unit disc, and show that the
cases where convergence is not uniform are very special indeed.
iii
Acknowledgements
Firstly I would like to thank my supervisor, Ian Short, for his extensive help, encourage-
ment, and boundless patience. I am very grateful for his support, without which I would
have made little progress – thank you! I would also like to thank Phil Rippon, my second
supervisor, for his generous support and advice. Everyone in the mathematics department
has been extremely welcoming, and so I thank them, most especially everyone in the com-
plex analysis group. I give particular thanks to the other Open University PhD students
for their friendship and support, especially David Bevan, David Martı-Pete, Grahame,
Mairi, Rosie and Vasso.
I would like to thank both examiners, Edward Crane (external) and David Brannan (in-
ternal), whose careful and extensive feedback on a previous version of this thesis has led
to an immensely improved final version.
I also give a huge thank-you to my mother and brother for their continuous support.
Finally, I would like to thank my partner, Anne-Marie, who encouraged me back into
maths and has endured much while I have pursued it. This thesis is dedicated to her.
v
Contents
Declaration i
Abstract iii
Acknowledgements v
Chapter 1. Introduction and background 1
1. Structure of this thesis 1
2. Continued fractions and composition sequences 2
3. The Mobius group and hyperbolic geometry 4
4. Sequences of Mobius transformations 9
5. Semigroups and Limit sets 10
Chapter 2. On Mobius semigroups 19
1. The group and inverse free parts of a semigroup 19
2. Semidiscrete semigroups 20
3. Schottky semigroups 21
4. Composition sequences 24
5. Covering regions 26
6. Constructing new semidiscrete semigroups from old 35
Chapter 3. Semigroups that fix the unit circle 39
1. A closer look at elementary subsemigroups of Aut(H2) 43
2. Proof of Theorem 3.1 46
3. Two-generator semigroups 49
4. Proof of Theorem 3.2 70
5. Classification of finitely-generated semigroups and proof of Theorem 3.3 78
vii
viii CONTENTS
6. Intersecting limit sets 83
Chapter 4. Limit sets 87
1. Introduction 87
2. On the structure of the limit sets 88
3. On semigroups with disjoint limit sets 105
Chapter 5. Self-maps of the disc 123
1. Introduction 123
2. Eventually tangent sequences of discs 129
3. Necessary and sufficient conditions for limit-disc type 130
4. Hausdorff dimension of the set of composition sequences of limit-disc type 133
5. Complete tangency graph 133
6. Convergence of composition sequences 141
Chapter 6. Coding limit sets 145
Bibliography 159
CHAPTER 1
Introduction and background
1. Structure of this thesis
In this first chapter we introduce the preliminary tools and ideas used in subsequent chap-
ters, as well as giving some historical background. We shall begin by giving the classical
definition of a continued fraction, which we shall later regard as a composition sequence
of Mobius transformations, before giving a brief overview of hyperbolic geometry. We
then introduce semigroups of Mobius transformations and discuss recent work of Fried,
Marotta and Stankewitz [15].
Chapter 2 builds on the theory of semigroups of Mobius transformations presented in
[15]. In Chapter 3 we study semidiscrete semigroups of Mobius transformations that fix
the unit disc. These can be regarded as semigroup analogues of Fuchsian groups. The
material in Chapters 2 and 3 is used in a joint paper [21] with Ian Short. Theorem 2.18 in
Chapter 2 is due to Edward Crane. In Chapter 4 we study the limit sets of semigroups in
detail. In Chapter 5 we turn to composition sequences generated by finitely many Mobius
transformations that each map the unit disc strictly within itself. The material in this
chapter is used in a joint paper [22] with Ian Short. Finally in Chapter 6 we consider what
can be regarded as a generalisation of the continued fraction expansion of a real number.
The work contained in Chapter 6 is in progress.
Throughout we shall freely assume the axiom of choice, which is used in, for example,
Konig’s lemma and the Baire category theorem.
1
2 1. INTRODUCTION AND BACKGROUND
2. Continued fractions and composition sequences
We denote the positive integers by N. A continued fraction is a formal expression
K(an|bn) =a1
b1 +a2
b2 + · · ·
,
where an and bn are infinite sequences of complex numbers and an 6= 0 for each n ∈ N.
We say that the continued fraction converges if the sequence with nth term
a1
b1 +a2
. . . +an
bn
converges in the extended complex plane C = C∪∞. We shall always consider continued
fractions as generated by infinite (rather than finite) sequences an and bn; in particular
we are excluding terminating continued fractions.
For a general choice of an and bn the continued fraction may not converge. It is well
known that if an = 1 and bn ∈ N for all n, then the continued fraction does converge.
Indeed, there is a one-to-one correspondence between each sequence b1, b2, . . . in N and
the irrational numbers contained in (0, 1) via the function that maps bn to the limit of the
sequence
1
b1 +1
. . . +1
bn
.
We equip the open unit ball B3 =
(x, y, z) ∈ R3 | x2 + y2 + z2 < 1
with the hyperbolic
metric, denoted ρ, which is induced by the line element
ds2 =4|dx|2
(1− |x|2)2,
where x is a point in R3. This means that, for points x and y in B3, the distance ρ(x,y)
is given by the infimum of the integral of this metric along all curves between x and y.
2. CONTINUED FRACTIONS AND COMPOSITION SEQUENCES 3
With respect to ρ there is a unique geodesic between any two points in B3. Every such
geodesic is an arc of a Euclidean circle orthogonal to the unit sphere, S2.
We define the ideal boundary of (B3, ρ) as follows. A geodesic half-ray is a map γ : [0,∞)→
B3 such that ρ(γ(0), γ(t)) = t for all t ∈ [0,∞). We define a relation ∼ on the set of ge-
odesic half-rays as follows. If γ1 and γ2 are geodesic half-rays, then γ1 ∼ γ2 if there is a
constant C > 0 such that ρ(γ1(t), γ2(t)) < C for all t. It is easy to check that ∼ is an
equivalence relation. The ideal boundary of (B3, ρ) is defined to be the set of equivalence
classes of geodesic half-rays, and can be identified with the unit sphere S2.
We consider the group Aut(B3) of Mobius transformations, defined to be the orientation-
preserving isometries of (B3, ρ). The action of Aut(B3) on B3 extends to a conformal action
on its boundary S2, which, by stereographic projection, we identify with the extended
complex plane C, as we explain shortly. It is well known that Aut(B3) is exactly the group
of conformal automorphisms of C. Moreover, when regarded as functions acting on the
complex plane, these Mobius transformations are exactly those maps of the form
z 7→ az + b
cz + d,
where a, b, c, d ∈ C and ad− bc = 1.
Given a collection F of Mobius transformations, we define a composition sequence gener-
ated by F to be a sequence
Fn = f1 · · · fn,
where each fi ∈ F . If fn(z) =an
z + bnfor each n, then the continued fraction K(an|bn)
converges exactly when Fn(0) converges. This is the connection between composition
sequences of Mobius transformations and continued fractions. In this thesis we use hy-
perbolic geometry, and other methods, to study composition sequences in detail. We are
particularly interested in the relationship between the collection of composition sequences
generated by some subset F of Aut(B3) and the semigroup generated by F .
4 1. INTRODUCTION AND BACKGROUND
Regarding continued fractions as composition sequences of Mobius transformations is use-
ful in at least two ways. Firstly, the tools of hyperbolic geometry can be applied directly to
the study of continued fractions, which can allow for the replacement of opaque algebraic
arguments with more concise geometric alternatives. Secondly, composition sequences give
a natural way of generalising continued fractions. For example, one can also consider com-
position sequences of Mobius transformations in higher dimensions. In the sequel we work
with Mobius transformations acting on B3 and S2, although many of our results also hold
for Mobius transformations acting on Bn and Sn−1. Following Aebischer’s 1990 paper [1],
several authors, notably Beardon, Hockman and Short (see for example [4]), have engaged
in a programme of interpreting and extending classical results from continued fraction
theory using the tools of hyperbolic geometry.
3. The Mobius group and hyperbolic geometry
We have already introduced Aut(B3) as the group of orientation-preserving isometries of
(B3, ρ); however, we now describe a more general collection of Mobius transformations.
We let R3 = R3 ∪ ∞ be the one-point compactification of R3. We denote the group of
conformal homeomorphisms of R3 by Aut(R3). The group Aut(R3) is the set of orientation-
preserving maps in the group that is generated by reflections in planes and inversions in
spheres acting on R3. The set Aut(B3) is the subgroup of elements in Aut(R3) that fix B3
setwise. We denote the distinguished point (0, 0, 0) in the unit ball by 0.
Occasionally it will be convenient to use a different model of hyperbolic space. We define
upper half-space by H3 =
(x, y, t) | (x, y) ∈ R2, t > 0
and denote the distinguished
point (0, 0, 1) ∈ H3 by j. The hyperbolic metric on H3, which we shall also denote by ρ, is
induced by the line element
ds2 =|dx|2
t2,
where x is a point in H3. Equipped with this metric, we call H3 the upper half-space
model. Exploiting the same construction used to define the ideal boundary of (B3, ρ), we
find that (H3, ρ) has a well-defined ideal boundary that we identify with C. Choose an
3. THE MOBIUS GROUP AND HYPERBOLIC GEOMETRY 5
element φ ∈ Aut(R3) that maps B3 onto H3, and maps the points 0, (0, 0, 1) and (0, 0,−1)
to j, ∞ and 0 respectively. Upon conjugating Aut(B3) by φ, we obtain the subgroup of
Aut(R3) that fixes H3. This is exactly the group of orientation-preserving isometries of
(H3, ρ).
We shall routinely switch between the upper half-space and unit ball models of hyperbolic
space, and do not make a notational distinction between a Mobius transformation acting
on one model and its corresponding conjugate acting on another. The upper half-space
model is usually preferred in an argument that considers a distinguished point on the ideal
boundary, which by conjugation we can take to be∞. On the other hand if our argument
involves a distinguished point in hyperbolic space, the unit ball model is useful so that,
by conjugation, this distinguished point may be taken to be 0.
The above discussion is dimensionless, although the number of conditions needed to iden-
tify a unique Mobius transformation from Hn onto Bn depends on n. In two dimensions,
we have the unit disc instead of the unit ball, which, when thought of as a subset of the
complex plane, carries the hyperbolic metric (also denoted ρ) given by the line element
ds2 =4|dz|2
(1− |z|2)2,
where z ∈ D. We call (D, ρ) the unit disc model of the hyperbolic plane. The group of
Mobius transformations acting on C that fix the unit disc is denoted Aut(D), and is exactly
the group of orientation-preserving isometries of (D, ρ). Choose a Mobius transformation
φ that maps the unit disc onto the upper half-plane H2 = (x, y) | y > 0, and maps 0 to
i. The map φ transfers the hyperbolic metric from the unit disc onto the upper half-plane.
In this model the hyperbolic metric is given by the line element
ds2 =|dz|2
Im(z)2,
where z ∈ H2. By conjugating Aut(D) by φ we obtain the collection of orientation-
preserving isometries of the upper half-plane.
6 1. INTRODUCTION AND BACKGROUND
Many of our results hold for Mobius transformations acting in higher dimensions. How-
ever, for clarity of exposition we shall always work with Mobius transformations acting on
at most three dimensions. If however a result is stated to hold for Mobius transformations
belonging to Aut(D) (or Aut(H2)), then we only claim the proof holds for these cases.
Many results in Chapter 3 are of this type.
We let χ denote the chordal metric on the closed unit ball, which is defined as the restriction
of the Euclidean metric on R3 to the closed unit ball. The unit sphere S2 can be identified
with C by the well-known stereographic projection map, which we now describe. A point
x ∈ S2 \ (0, 0, 1) is first mapped to (x, y, 0) ∈ R3, defined as the point where the plane(x, y, z) ∈ R3 | z = 0
intersects the infinite line in R3 that passes though x and (0, 0, 1).
The stereographic projection map sends x ∈ S2 \ (0, 0, 1) to x+ iy ∈ C, while the point
(0, 0, 1) is mapped to ∞ ∈ C. If z and w are points in C then χ(z, w) is given by
(1) χ(z, w) =2|z − w|√
1 + |z|2√
1 + |w|2,
where it is understood that the appropriate limit is taken if one or both of z or w equals
∞. We now describe a complete metric on the group Aut(B3) itself. For f, g ∈ Aut(B3),
the metric of uniform convergence is given by
(2) σ(f, g) = supz∈B3
|f(z)− g(z)|.
Directly from its definition, σ can be seen to be right invariant, that is
σ(fh, gh) = σ(f, g)
for all f, g, h ∈ Aut(B3). In fact Aut(B3) is a topological group [4, Theorem 3.1], that is,
the maps f 7→ f−1 from Aut(B3) to itself, and (f, g) 7→ fg from Aut(B3) × Aut(B3) to
Aut(B3) are both continuous.
The map Φ given by
Φ
a b
c d
=
(z 7→ az + b
cz + d
)
3. THE MOBIUS GROUP AND HYPERBOLIC GEOMETRY 7
from SL(2,C) onto Aut(B3) is a surjective group homomorphism with kernel I,−I,
where I is the identity matrix. This implies that Aut(B3) is isomorphic to PSL(2,C) =
SL(2,C)/ I,−I.
For any Mobius transformation f the square of the trace of both representations of f
in SL(2,C) is the same. This implies that the function tr2 : Aut(B3) → C given by
tr2(f) = (trace(F ))2, where F ∈ SL(2,C) satisfies Φ(F ) = f , is well defined. Indeed, Φ is
also an open, continuous and surjective map [4, Theorem 3.2].
The function tr2 is important because two elements f, g ∈ Aut(B3) are conjugate if and
only if tr2(f) = tr2(g) (see [4, Theorem 3.10]). We let I denote the identity map. A
Mobius transformation f not equal to the identity falls into one of the following three
classes:
• The map f is parabolic if f is conjugate to the map z 7→ z + 1.
In this case tr2(f) = 4 and f has exactly one fixed point in C with multiplier
equal to 1.
• The map f is loxodromic if f is conjugate to a map of the form z 7→ λz where
|λ| 6= 1.
In this case tr2(f) ∈ C \ [0, 4] and f has exactly two fixed points in C, one
attracting and one repelling.
• The map f is elliptic if f is conjugate to a map of the form z 7→ λz where |λ| = 1
and λ 6= 1.
In this case tr2(f) ∈ [0, 4) and f has exactly two fixed points in C, each with
multiplier of modulus 1.
It is useful to observe that the function tr2 is continuous. Since Φ is an open map, it
follows that the collection of loxodromic elements is an open subset of Aut(B3).
If a Mobius transformation g satisfies gn = I for some positive integer n, then we call the
least such n the order of g, and write n = order(g). If there is no such positive integer n,
then we say that g has infinite order. By the characterisation of Mobius transformations
given above, it is clear that only elliptic transformations conjugate to rational rotations
8 1. INTRODUCTION AND BACKGROUND
have finite order. All other nontrivial Mobius transformations have infinite order.
Let us now give a geometric description of the action of each type of Mobius transformation
on hyperbolic space. Each elliptic and loxodromic transformation has two fixed points on
the ideal boundary. The hyperbolic geodesic landing at these fixed points is called the axis
of the transformation. Each elliptic transformation fixes its axis pointwise. The action
of the elliptic transformation is to perform a nontrivial hyperbolic rotation around its
axis. The angle of rotation, θ, of an elliptic transformation f is related to tr2(f) by the
equation tr2(f) = 4 cos2(θ/2). A loxodromic transformation f has one attracting fixed
point and one repelling fixed point, which we shall denote by αf and βf respectively. The
map f first translates each point x in B3 by some nonzero hyperbolic distance, called
the translation length of f , parallel to its axis and in the direction of the attracting fixed
point of f , before performing a (possibly trivial) hyperbolic rotation about its axis. A
loxodromic transformation fixes its axis setwise. The action of a parabolic transformation
is very different. A horoball is a Euclidean sphere internally tangent (and not equal) to the
unit sphere, and we call the point of tangency the base point of the horoball. A parabolic
transformation fixes any horoball based at its fixed point setwise. The action of a parabolic
element f is best visualised by thinking in the upper half-space model, and conjugating
f to the map z 7→ z + 1, which fixes the horoball
(x, y, t) | (x, y) ∈ R2 ∪ ∞
for any
given t > 0.
To understand the action of a Mobius transformation on the ideal boundary, the notion
of isometric discs can be helpful. First, we define the chordal derivative of a Mobius
transformation f at w ∈ C to be the quantity
limz→w
χ(f(z), f(w))
χ(z, w).
For any Mobius transformation f , its isometric disc, which we shall denote by I+(f), is
the open chordal disc contained in C where the chordal derivative of f is greater than 1.
We let I−(f) denote the isometric disc of f−1. If f is loxodromic then αf ∈ I−(f) and
βf ∈ I+(f). Both isometric discs have chordal radius 1/ sinh[12ρ(f(0),0)]. The transfor-
mation f maps the interior of the complement of I+(f) onto I−(f). See [29, Chapter
1] for more details on the action of Mobius transformations on hyperbolic space and the
4. SEQUENCES OF MOBIUS TRANSFORMATIONS 9
properties of isometric discs.
Definition A Kleinian group is a discrete group of Mobius transformations. A Fuchsian
group is a Kleinian group whose elements all fix some non-trivial chordal disc in C. Equiv-
alently, a Fuchsian group is a Kleinian group that is conjugate to a discrete subgroup of
Aut(D) by a Mobius transformation.
We recall the following definition of a Schottky group, which may be found in [29, Sec-
tion 2.7].
Definition Let C1, C′1, . . . , Ck, C
′k be k pairs of circles in C, whose interiors are pairwise
disjoint. Suppose that for each n = 1, . . . , k the Mobius transformation gn maps Cn
onto C ′n, and maps the interior of Cn onto the exterior of C ′n. The group generated by
g1, . . . , gn is a discrete group. We say that a group of Mobius transformations is a
Schottky group if it is conjugate to a group constructed as above.
For some authors the definition given above is that of a classical Schottky group, and
they give a more general definition of a Schottky group, where the circles featured in the
construction are replaced by Jordan curves.
Schottky groups are examples of free groups. A free group is a group that has no non-trivial
relations. We shall also need the related but slightly different notion of a free semigroup.
We shall say that a semigroup S generated by a set F of Mobius transformations is free if
each element in S can be written uniquely as a finite composition of elements in F \ I.
4. Sequences of Mobius transformations
A sequence of Mobius transformations gn is said to converge ideally if for any point ζ ∈ B3
the sequence gn(ζ) converges in the Euclidean metric to a point on the ideal boundary.
This definition is independent of the choice of point ζ in hyperbolic space. We say ‘con-
verges ideally’ rather than simply ‘converges’, as the latter term is reserved to describe the
convergence of gn as a sequence inside the Mobius group, that is, if gn converges uniformly
to a Mobius transformation. Ideal convergence was introduced to the theory of continued
10 1. INTRODUCTION AND BACKGROUND
fractions by Lorentzen (formerly known as Jacobson) in [26], although there the term gen-
eral convergence was used, and it was defined purely in terms of the action of Aut(B3) on C.
The notion of an ideally converging sequence of Mobius transformations will play an
important role in this thesis. It is useful to define another, weaker property on sequences
of Mobius transformations. We say that a sequence gn in Aut(B3) is escaping if for
any ζ ∈ B3 the sequence gn(ζ) accumulates only on the boundary of hyperbolic space,
with respect to the Euclidean metric on B3. If a sequence gn is escaping then we call
the sequence an escaping sequence. Again, since each gn is a hyperbolic isometry, this
definition is independent of the choice of ζ. In the continued fractions literature, what
we call escaping sequences are sometimes called ‘restrained sequences’. These were first
defined (see [28, Definition 2.6]) by Lorentzen and Thron purely in terms of the action of
a sequence on C.
5. Semigroups and Limit sets
We define a semigroup of Mobius transformations, or more briefly, a semigroup, to be a
set of Mobius transformations that is closed under composition. Note that the identity
element, which we shall denote by I, need not lie in a semigroup.
Throughout the thesis, we write A ⊆ B to mean that A is contained in or equal to B.
If A is contained in but not equal to B, then we shall write A ( B. For any collection
X ⊆ Aut(B3) of Mobius transformations and any ζ ∈ B3, we describe the set of points
X−1(ζ) =g−1(ζ) | g ∈ X
as the backward orbit of ζ under X. Similarly, X(ζ) =
g(ζ) | g ∈ X is the forward orbit of ζ under X, or the X-orbit of ζ. We now introduce
one of the central ideas within this thesis.
Definition Choose any point ζ ∈ B3. We define the backward limit set of X as
Λ−(X) = X−1(ζ) ∩ S2,
where the closure is with respect to the Euclidean metric. Similarly we define the forward
limit set of X as
Λ+(X) = X(ζ) ∩ S2.
5. SEMIGROUPS AND LIMIT SETS 11
These definitions are independent of the choice of ζ in hyperbolic space. Both limit sets
are closed subsets of S2. If gn is a sequence then we shall refer to Λ−(gn | n ∈ N) as the
backward limit set of gn, and abbreviate this simply to Λ−(gn). Similarly we write Λ+(gn)
for Λ+(gn | n ∈ N) and call this the forward limit set of gn. When X is a Kleinian group,
the forward and backward limit sets coincide with the limit set of the Kleinian group. A
good introduction to Kleinian groups and the properties of their limit sets can be found
in [3] or [29]. We shall be interested in limit sets where X is either a semigroup, or a
sequence of Mobius transformations. Indeed, the limit sets of a semigroup will be of great
interest to us, and we shall study them in detail. Limit sets of semigroups were studied
by Fried, Marotta and Stankewitz in [15], and we shall develop their ideas. Limit sets
of sequences are important because of the following theorem due to Aebischer [1], which,
for a sequence of Mobius transformations gn, connects convergence of the sequence gn(z)
where z is a point on the ideal boundary, to convergence of gn(ζ) where ζ is a point in
hyperbolic space. See also [4, Theorem 9.6].
Theorem 1.1. Suppose that gn is an escaping sequence of Mobius transformations. Then
χ(gn(z), gn(0)) → 0 as n → ∞ for all z ∈ B3 \ Λ−(gn). In fact convergence is locally
uniform in B3 \ Λ−(gn), and B3 \ Λ−(gn) is the largest open set with this property.
In fact, it is shown in [1] that for any X the complement of Λ−(X) is the largest open
subset of the ideal boundary upon which X is a normal family.
We now define two important subsets of the backward and forward limit sets, which have
familiar counterparts in the theory of Kleinian groups. Suppose that X is a set of Mobius
transformations and let z ∈ S2. Choose any hyperbolic geodesic segment γ that has one
end point z and the other end point in B3, and let ζ ∈ B3. Then z is a forward conical
limit point of X if there is sequence gn in X, and a positive constant M , such that
(i) gn(ζ)→ z as n→∞;
(ii) ρ(γ, gn(ζ)) < M for all n.
For such a sequence gn, we shall say that gn(ζ) converges conically to z. These definitions
do not depend on the choice of γ or ζ. The forward conical limit set Λ+c (X) of X is the
collection of all forward conical limit points of X. A backward conical limit point of X
is defined to be a forward conical limit point of X−1, and the backward conical limit set
12 1. INTRODUCTION AND BACKGROUND
Λ−c (X) of X is defined to be Λ+c (X−1).
We also define the forward horospherical limit set Λ+h (X) of X to consist of those points
z in Λ+(X) such that, for any point ζ in B3, the orbit X(ζ) meets every horoball based
at z. The backward horospherical limit set Λ−h (X) of X is defined as Λ+h (X−1). Clearly
we have the inclusions
Λ+c ⊆ Λ+
h ⊆ Λ+ and Λ−c ⊆ Λ−h ⊆ Λ−.
Here we have omitted mention of X, as we often will do for brevity. Conical and horospher-
ical limit sets have important roles in the theory of Kleinian groups, so it is no surprise
that the sets introduced here play an important part in the study of semigroups.
Given a semigroup S, we say that a subset Y of B3 is forward invariant (with respect
to S) if for each element g of S, g(Y ) ⊆ Y . Similarly we shall say that Y ⊆ B3 is
backward invariant if for each element g of S, g−1(Y ) ⊆ Y . Equivalently, Y is backward
invariant with respect to S if Y is forward invariant with respect to S−1. In [15] the
authors make the important observation that Λ+(S) is forward invariant and Λ−(S) is
backward invariant. Indeed, the conical and horospherical limit sets behave in the same
way: Λ+c (S) and Λ+
h (S) are forward invariant, while Λ−c (S) and Λ−h (S) are backward in-
variant, which one can verify directly from the definitions.
The backward conical limit set of sequences features in the theory of continued fractions
because of the following theorem of Aebischer [1, Theorem 5.2].
Theorem 1.2. Let gn be an escaping sequence, and let ζ ∈ B3. Then, for each point z
in S2, we have χ(gn(ζ), gn(z))→ 0 as n→∞ if and only if z /∈ Λ−c (gn).
It is straightforward to prove this theorem using hyperbolic geometry; see, for example,
[11, Proposition 3.3]. We also have the following lemma, proven in [11, Lemma 3.4].
Lemma 1.3. Suppose that w is a backward conical limit point of an escaping sequence gn.
Then there is a sequence of positive integers n1, n2, . . . and two distinct points p and q in
S2 such that gni(w)→ p and gni(z)→ q for every point z in S2 other than w.
5. SEMIGROUPS AND LIMIT SETS 13
Clearly any group is a semigroup with equal backward and forward limit sets. In general
the limit sets of a semigroup do not coincide, and we shall study how the two sets relate
to each other and what they can tell us about the underlying semigroup.
For any set F ⊆ Aut(B3) we shall write 〈F〉 to denote the semigroup consisting of all finite
compositions of elements that belong to F . We shall call 〈F〉 the semigroup generated by
F . If S = 〈F〉 for some finite collection F of Mobius transformations, then we say that
S is finitely-generated. To be concise we denote the semigroup generated by elements
f, g ∈ Aut(B3) by 〈f, g〉, rather than by 〈f, g〉.
It is often convenient to conjugate a semigroup by a Mobius transformation. One reason
this is useful is because, as is well known, the group of Mobius transformations is transitive
on distinct ordered triples. That is, for any two ordered lists of three distinct points in C,
(p, q, r) and (p′, q′, r′) say, there is a unique Mobius transformation that maps p, q and r to
p′, q′ and r′ respectively. If S is a semigroup then we can conjugate S by any f ∈ Aut(B3)
to give another semigroup
T = fSf−1 =fgf−1 | g ∈ S
.
It is clear that Λ+(T ) = f(Λ+(S)) and Λ−(T ) = f(Λ−(S)). Throughout this thesis when-
ever we say that a semigroup S is conjugate to a semigroup T we shall mean conjugate
by an element in Aut(B3). If S and T are semigroups and S ⊆ T , then we shall describe
S as a subsemigroup of T . Finally we observe that if S is a semigroup, then so is S−1.
In [15], Fried, Marotta and Stankewitz introduce semigroups of Mobius transformations
and backward and forward limit sets. They go on to show that if the backward and forward
limit sets of a semigroup are disjoint, then the semigroup can be regarded as an iterated
function system on a certain metric space, which they construct. Recall that the limit set
Λ(G) of a Kleinian group G is either a perfect set, or contains 0, 1 or 2 points. Moreover,
a group G is said to be elementary if Λ(G) is not a perfect set, or equivalently, if there
exists a point x ∈ B3 such that the G-orbit of x is finite. The following theorem, given in
14 1. INTRODUCTION AND BACKGROUND
[15, Theorem 2.11] shows that just as for groups, semigroups of Mobius transformations
come in varieties that can be regarded as elementary and nonelementary.
Theorem 1.4. Let S be a semigroup. Then either Λ+(S) is a perfect set or S is conjugate
in Aut(B3) to a semigroup T of one of the following types.
(i) |Λ+(T )| = 0 and T is a semigroup of elliptic maps with a common fixed point,
(ii) |Λ+(T )| = 1 and each g ∈ T is of the form g(z) = az+ b with a, b ∈ C and |a| ≥ 1, or
(iii) |Λ+(T )| = 2 and each g ∈ T is of the form g(z) = az or g(z) = a/z for some
a ∈ C \ 0.
Definition We say that a semigroup S is elementary if there is a point z ∈ B3 for which
S(z) is finite.
This definition is at odds with the definition used in [15], where a semigroup is described
as elementary whenever Λ−(S) is finite. The definition used in [15] has two undesirable
consequences. Firstly, it is possible for S to be elementary and S−1 not elementary; sec-
ond, with this definition and when S happens to be a group, it is possible for S to be an
elementary group but not an elementary semigroup. The former case happens if S = 〈f, g〉
where f(z) = 13z and g(z) = 1
3(z + 2). Then Λ+(S) is equal to the middle-thirds Cantor
set and Λ−(S) = ∞. The latter case, where S an elementary group but Λ−(S) is not
finite, happens if, for example, S is the stabilizer in Aut(B3) of some point z ∈ C. We
shall call 〈f, g〉 the Cantor semigroup. The Cantor semigroup, which is given as an exam-
ple in [15], is familiar and easy to analyse, and we shall use it again to illustrate new ideas.
The elementary semigroups are classified in the following theorem.
Theorem 1.5. Let S be an elementary semigroup. Then either
(i) there is a point in B3 fixed by all elements of S; or
(ii) there is a point in C fixed by all elements of S; or
(iii) there is a pair of points in C fixed setwise by all elements of S.
Proof. Let S be an elementary semigroup with finite orbit X. We can assume that
X is either contained in B3 or its ideal boundary. Any finite orbit is forward invariant,
and so must be fixed setwise by each element in S.
5. SEMIGROUPS AND LIMIT SETS 15
One possibility is that |X| ≥ 3 and X is contained in S2. Since an element in Aut(B3) is
uniquely determined by its action on three distinct points in S2, S itself must be finite, and
in particular for any point ζ ∈ B3 its orbit S(ζ) is finite. In this case we write K = S(ζ).
Another possibility is that X ⊆ B3, in which case we set K equal to X. In both these cases
we have a finite set K ⊆ B3 that is fixed setwise by each element in S. One possibility
is that K is a single point. Otherwise we can define B to be the unique smallest closed
hyperbolic ball containing K. We claim that S fixes the (hyperbolic) centre of B. For any
g ∈ S we have K ⊆ B ∩ g(B). If g does not fix B, then, by conjugating so that the origin
0 is the midpoint of the line from the centre of B to the centre of g(B), it can be seen
that B ∩ g(B) is contained in a hyperbolic ball of radius strictly less than the radius of B,
contrary to our choice of B. It follows that no such g exists, as claimed. Hence if either
X ⊆ B3, or both |X| ≥ 3 and X ⊆ C, then we must have case (i). It remains to consider
the case where |X| ≤ 2 and X ⊆ C. We have already shown that each element of S fixes
X setwise, hence we have case (ii) if |X| = 1 and case (iii) if |X| = 2.
The next observation follows immediately from the theorem.
Corollary 1.6. The semigroup S is elementary if and only if S−1 is elementary.
Below we give a finer classification of elementary semigroups, which follows easily from
Theorems 1.4 and 1.5, and whose proof we omit.
Proposition 1.7. Any nontrivial elementary semigroup S is of one of the following types:
(a) Λ+ = Λ− = ∅ and S is a semigroup of hyperbolic rotations about a point in hyperbolic
space.
(b) Λ+ = Λ− = x and S contains parabolic maps fixing x and possibly elliptic maps
fixing x. These are the only possible maps in S.
(c) Λ+ = α and Λ− = β where α 6= β. S contains one or more loxodromic maps with
attracting fixed point α and repelling fixed point β. There may also be elliptic maps
that fix both points.
(d) |Λ+| = 1 and |Λ−| > 2. If Λ+ = α then S contains loxodromic maps with attracting
fixed point α and possibly parabolic or elliptic maps fixing α. The semigroup must
contain at least two such loxodromic maps with different repelling fixed points.
16 1. INTRODUCTION AND BACKGROUND
(e) |Λ−| = 1 and |Λ+| > 2. Here the inverse semigroup S−1 satisfies the preceding case.
(f) |Λ+| = |Λ−| = 2. In this case Λ+ = Λ− = x, y and S contains loxodromic maps and
possibly elliptic maps fixing x, y setwise. Both x and y must belong to both the set
of attracting fixed points and the set of repelling fixed points of S.
(g) For some z ∈ C each element in S fixes z. Moreover |Λ+| > 2 and |Λ−| > 2.
In particular a semigroup that has at least one finite limit set is necessarily elementary.
Case (g) represents the only elementary semigroups for which both limit sets are infinite.
The distinction between elementary semigroups that have at least one finite limit set and
those in case (g) is of significance. Accordingly we make the following definition.
Definition We say that an elementary semigroup S is of finite type if at least one of
Λ+(S) and Λ−(S) is a finite set. An elementary semigroup that is not of finite type is of
infinite type.
In Chapter 3 we shall consider subsemigroups of Aut(D), and there we give a specialised
version of the above proposition, so that we can more precisely understand such semi-
groups. Notice that if one limit set contains exactly 0 or 2 points, then so does the other
limit set. Cases (d) and (e) are the only classes of elementary semigroup for which exactly
one of the limit sets is perfect. If both limit sets are singletons, then they may or may not
be equal. Case (a), where both limit sets are empty, is exactly case (i) in Theorem 1.5.
Cases (c) and (f) correspond to case (iii) in Theorem 1.5. All other cases ((b),(d),(e), and
(g)) correspond to case (ii) in Theorem 1.5.
An example of case (e) is given by the Cantor semigroup, that is the semigroup 〈f, g〉
where f(z) = 13z and g(z) = 1
3(z + 2).
In the theory of Kleinian groups, the limit set Λ(G) of a Kleinian group G is the set of
non-normality of G, that is, the complement of Λ(G) is the largest open set upon which
G is a normal family. Similarly if f is an analytic self-map of a Riemann surface, then
the Julia set J(f), defined to be the set of non-normality of the family fn | n ∈ N, is
the smallest closed set that is fixed setwise by f . Moreover, when G is a nonelementary
Kleinian group, the repelling fixed points of G are dense in Λ(G), and, provided f is not
5. SEMIGROUPS AND LIMIT SETS 17
a parabolic Mobius transformation, the repelling fixed points of fn | n ∈ N are dense
in J(f). For semigroups of Mobius transformations we have the following result.
Theorem 1.8. Suppose that S is a Mobius semigroup. Then:
(i) provided S contains loxodromic elements, the set of repelling fixed points of S is dense
in Λ−;
(ii) the complement of Λ− in S2 is the largest open set upon which S is a normal family;
and
(iii) if Λ− contains at least three points then it is the smallest closed, backward invariant
subset of the ideal boundary containing at least three points.
Proof. The first statement is given in [15, Theorem 2.4]. The second was essentially
first proven in [1, Theorem 3.3], but also appears as [15, Theorem 2.6]. The final statement
is [15, Remark 2.20]. The conclusion of statement (i) holds for all semigroups except those
elementary semigroups that are not of case (b) in Proposition 1.7.
By applying Theorem 1.8 to the nonelementary semigroup S−1, we observe that the at-
tracting fixed points of S are dense in Λ+. Moreover, Λ+ is the smallest closed, forward
invariant subset of S2 containing at least three points. It follows that any forward invariant
set containing at least three points is dense in Λ+, so that in particular Λ+c and Λ+
h are
dense in Λ+.
The concept of non-normality can be used to give a unified definition of the limit set Λ(G)
of a Kleinian group G and the Julia J(f) set of an analytic function f . The above theorem
tells us that we can also define the backward limit set Λ− of a Mobius semigroup using
normality. There are, however, important differences between the behaviour of Λ− and
of Λ(G) and J(f). The set J(f) is fixed setwise by f , and Λ(G) is fixed setwise by any
element of G. In contrast for a semigroup S, in general neither limit set is fixed by all the
elements in S.
We finish this chapter with some remarks on the dynamics of semigroups of analytic
functions under composition. The study of semigroups of rational functions was introduced
18 1. INTRODUCTION AND BACKGROUND
in 1995 by Hinkkanen and Martin [20]. They define the Julia set of a semigroup of
rational functions acting on C to be the set of non-normality of the semigroup. In [20]
the semigroup is taken to include at least one rational function that is not a Mobius
transformation. The subject of semigroups of rational functions has since been further
developed (see for example [19, 36–39]). Semigroups of transcendental entire functions
acting on C have also been studied; see, for example, [24].
CHAPTER 2
On Mobius semigroups
In this chapter we develop the theory of semigroups of Mobius transformations. In par-
ticular, we introduce the idea of a semidiscrete semigroup, which can be regarded as a
generalisation of a discrete group. Some of our results on semidiscrete semigroups are
analogues of known results on discrete groups, and it appears that the semidiscreteness
property plays a role for semigroups somewhat similar to that of the discreteness property
for groups.
1. The group and inverse free parts of a semigroup
The group part of a semigroup S is the set of elements in S whose inverses also lie in S.
Hence the group part of S is the largest subset of S that is also a group, and is equal to
S ∩ S−1. We call the complement of S ∩ S−1 in S the inverse free part of S, as it does
not contain the identity. One possibility is that S itself is a group. In the other extreme,
the group part of S might be empty, or equivalently I /∈ S. This partition of a semigroup
into its group part and inverse free part is particularly important because of the following
lemma.
Lemma 2.1. Let S be a semigroup. If g, h ∈ S, and one of them belongs to S \ S−1, then
gh ∈ S \ S−1.
Proof. We prove the contrapositive assertion that if gh ∈ S∩S−1 then g, h ∈ S∩S−1.
If gh ∈ S ∩ S−1, then f = (gh)−1 is an element of S. Therefore g−1 = hf and h−1 = fg
both belong to S, as required.
It follows from Lemma 2.1 that the inverse free part of a semigroup is itself a semigroup.
We define a word generated by a collection F of Mobius transformations to be an ex-
pression of the form f1 . . . fn, where n ∈ N and each fi belongs to F . By regarding each
19
20 2. ON MOBIUS SEMIGROUPS
word as a composition of Mobius transformations, each word can be mapped to a Mobius
transformation belonging to the semigroup S generated by F . This mapping is certainly
surjective. We say S is freely generated by F (or simply free) if the mapping is also
injective.
Lemma 2.2. Suppose that S is a semigroup generated by a set F . If S has a nonempty
group part, then that group is generated by a subset of F .
Proof. Let G denote the set of elements in F that belong to the group part of S. By
Lemma 2.1, a word in the generators F represents an element of S ∩S−1 if and only if all
the letters in the word belong to G. Therefore S ∩ S−1 is generated by G.
In particular, the group part of a finitely-generated semigroup S is finitely-generated.
In contrast, the inverse free part of S, that is, S \ S−1, need not be finitely-generated,
as the following example illustrates. Let f and g be two Mobius transformations that,
as a group, generate a Schottky group. Consider the semigroup S =⟨f, f−1, g
⟩. The
set S \ S−1 is exactly those Mobius transformations that arise from words generated by
the set f, f−1, g that feature at least one occurrence of g. Since the group generated
by f and g is freely generated, any finite set in S \ S−1 cannot generate gfn for every
n ∈ N. This means that even though S is finitely-generated, S \ S−1 is not. In Section
2.4 of this chapter we shall consider how the limit sets of S\S−1 relate to the limit sets of S.
2. Semidiscrete semigroups
We now introduce an important property of semigroups.
Definition We say that a set X of Mobius transformations is semidiscrete if the identity
element I is not an accumulation point of X in Aut(B3).
As an example, consider the semigroup S generated by f(z) = 2z and g(z) = 12z+1. These
are loxodromic Mobius transformations such that the attracting fixed point of f equals
the repelling fixed point of g, that is αf = βg. This semigroup is not discrete because
gnfn(z) = z + 2− 1
2n−1→ z + 2 as n→∞.
3. SCHOTTKY SEMIGROUPS 21
However, it is semidiscrete because each element of S has the form z 7→ 2nz + b, where
n ∈ Z and b ∈ R, and it is easy to check that if n = 0 then b > 1.
In the theory of groups there are several properties that are equivalent to discreteness.
Accordingly, we now consider properties of semigroups that are equivalent to being semidis-
crete. The action of a semigroup S on B3 is said to be properly discontinuous if for each
point ζ in B3 there is a neighbourhood U of ζ such that g(U) ∩ U 6= ∅ for only finitely
many elements g of S. When S is a group, it is discrete if and only if its action on B3
is properly discontinuous, and this is so precisely when the S-orbit of any point in B3 is
locally finite. The next theorem is a comparable result for semigroups.
Theorem 2.3. Let S be a semigroup. The following statements are equivalent:
(i) S is semidiscrete;
(ii) the action of S on B3 is properly discontinuous;
(iii) the S-orbit of any point ζ in B3 does not accumulate at ζ.
We omit the elementary proof, which is similar to proofs of analogous theorems from the
theory of Kleinian groups.
The action of a semigroup S on B3 is said to be strongly discontinuous if for each point ζ
in B3 there is a neighbourhood U of ζ such that g(U) ∩ U = ∅ for every element g of S.
This definition is close to the definition of a discontinuous action in the theory of Fuchsian
groups, but there the intersection g(U) ∩ U is only required to be empty for elements of
the group other than the identity.
Theorem 2.4. Let S be a semigroup. The following statements are equivalent:
(i) S is semidiscrete and inverse free;
(ii) the action of S on B3 is strongly discontinuous;
(iii) the S-orbit of any point ζ in B3 stays a positive distance away from ζ.
Again, the proof is straightforward, and omitted.
3. Schottky semigroups
Here we introduce a large class of finitely-generated, inverse free semidiscrete semigroups.
We first give a condition which ensures that a semigroup is inverse free and semidiscrete.
22 2. ON MOBIUS SEMIGROUPS
Theorem 2.5. Let K be a closed subset of S2 and let F be a finite subset of Aut(B3)
comprised of transformations that map K within itself. Further suppose that the elements
of F that fix K setwise generate a finite group. Then the semigroup S generated by F is
semidiscrete. If no element in F fixes K setwise, then S is also inverse free.
Proof. First we prove that the semigroup S generated by F is semidiscrete. Suppose,
on the contrary, that S is not semidiscrete; then there is a sequence of distinct elements
gn of S that converges to the identity map I. Let F0 denote those elements in F that
fix K setwise, and let F1 denote those elements in F that map K strictly within itself.
Let G be the group generated by F0. Each map gn can be written as a composition of
elements of F . One possibility is that infinitely many gn lie in G. Then since G is fi-
nite we have gn = I for infinitely many n, contrary to our assumption that the elements
in the sequence gn are distinct. Otherwise we may pass to a subsequence such that we
can write gn = pfqn, for n = 1, 2, . . . , where f ∈ F1, qn ∈ S and p ∈ G ∪ I. Then
gn(K) = p(f(qn(K))) ⊆ p(f(K)). Since f(K) is a proper subset of K, so is pf(K). It
follows that gn cannot accumulate at the identity. Hence S is semidiscrete.
To show the contrapositive of the final statement, suppose that S contains the identity.
Then I = f1 · · · fm for some f1, . . . , fm ∈ F and positive integer m. Suppose towards
contradiction that some fi lies in F1. Then f1 · · · fm(K) ⊆ f1 · · · fi(K) ( f1 · · · fi−1(K),
so that f1 · · · fm cannot be the identity map. Hence each fi lies in F0, and in particular
F0 is nonempty.
Suppose now that K is the union of a finite collection of disjoint, closed discs in S2, and
let F be a finite subset of Aut(B3) made up of transformations that map K strictly within
itself. A Schottky semigroup is a semigroup generated by such a set F . We use this termi-
nology because Schottky groups contain many Schottky semigroups; for example, if f and
g generate as a group a Schottky group, then f and g generate as a semigroup a Schottky
semigroup.
We now give an example of a finitely-generated inverse free semidiscrete semigroup that
is not a Schottky semigroup. In this example, as usual, we denote the attracting and
3. SCHOTTKY SEMIGROUPS 23
repelling fixed points of a loxodromic element g of Aut(B3) by αg and βg, respectively.
Let f , g and h be loxodromic maps in Aut(D) that generate as a group a Schottky group
(not a Schottky semigroup). Choose these maps in such a way that αf and αh lie in
different components of S1 \ αg, βg. Let q = fg−1f−1, and define S = 〈f, g, h, q〉 (see
Figure 2.1). This semigroup lies in a discrete group, so it is semidiscrete. To see that S is
inverse free, suppose, in order to reach a contradiction, that w1 · · ·wn = I, where wi is a
positive power of either f , g, h or q for i = 1, . . . , n. By thinking of w1 · · ·wn as a word in
f , g and h, we see that wi cannot equal f or h (because the sum of the powers of each of
f , g and h in this word must be 0, as those three maps generate a free group). Therefore
each map wi is equal to either gm or fg−mf−1, for some positive integer m, so clearly it
is not possible for w1 · · ·wn to equal the identity after all.
It remains to prove that there is not a finite collection of disjoint, closed intervals in S1
whose union K is mapped strictly within itself by each element of S. Suppose there is
such a set K. Then it must contain Λ+(S), so, in particular, it contains αg. Furthermore,
it contains gn(αf ) and gn(αh) for each positive integer n. These points accumulate on
either side of αg (our initial choice of f , g and h ensures that this is so). It follows that
αg is an interior point of K. Now, q = fg−1f−1, so βq = f(αg), which implies that βq is
also an interior point of K. However, this is impossible because q(K) ⊆ K. Therefore S
is not a Schottky semigroup.
f
g
h
q
Figure 2.1. The axes of f, g, q, h
An important feature of this example is that the generating set is not unique, because
S = 〈fg, g, h, q〉, as one can easily verify. Although S is contained in a Schottky group,
24 2. ON MOBIUS SEMIGROUPS
this property is not mandatory for a finitely-generated, semidiscrete and inverse free semi-
group whose generating set is not unique. Indeed, by appending an appropriate Mobius
transformation onto S, we can generate a larger semigroup with these same properties that
is not contained in a Schottky group. More specifically, we can use the construction out-
lined in Chapter 2, Section 6 to choose an element t ∈ Aut(D) such that T = 〈f, g, h, q, t〉
is also semidiscrete, inverse free and finitely-generated, but not uniquely generated by
f, g, h, q, t, nor contained within a discrete group. The problem of determining whether
or not a particular generating set for a semigroup is unique will be dealt with in Theo-
rem 4.25 and the surrounding discussion.
4. Composition sequences
Recall that a sequence Gn of Mobius transformations is an escaping sequence if, for some
point ζ in B3, the orbit Gn(ζ) does not accumulate at any point in B3. There are various
equivalent ways of describing escaping sequences, one of which is captured in the following
standard lemma (we omit the elementary proof).
Lemma 2.6. A sequence of Mobius transformations is an escaping sequence if and only if
it does not contain a subsequence that converges to a Mobius transformation.
The statement that a sequence Gn of Mobius transformations converges uniformly to
another Mobius transformation G is equivalent to σ(Gn, G) → 0 as n → ∞, where σ is
the metric of uniform convergence introduced in Chapter 1. The relationship between
composition sequences and semigroups is a central theme of this thesis. The key to this
relationship is the following simple theorem, which characterises when every composition
sequence generated by S converges.
Theorem 2.7. Let S = 〈F〉 be a semigroup. The following are equivalent:
(i) S is semidiscrete and inverse free.
(ii) Every composition sequence generated by F is an escaping sequence.
(iii) Every composition sequence generated by S is an escaping sequence.
Proof. We first show that (iii) implies (i). Suppose that S is not both semidiscrete
and inverse free. Then either the identity transformation I is an accumulation point of
4. COMPOSITION SEQUENCES 25
S in Aut(B3), or else it belongs to S. In other words, I ∈ S. Hence there is a sequence
gn in S that converges to I. By restricting to a subsequence of gn, we can assume that∑σ(gn, I) < +∞. Now define Gn = g1 · · · gn for n ∈ N. Using the right-invariance of σ
we see that
σ(G−1n , G−1n−1) = σ(G−1n Gn, G−1n−1Gn) = σ(I, gn).
Therefore∑σ(G−1n , G−1n−1) < +∞, which implies that G−1n is a Cauchy sequence. Hence
Gn is a Cauchy sequence too. As σ is a complete metric on Aut(B3), the sequence Gn
converges. Hence by Lemma 2.6, Gn is not an escaping sequence, and so we have found a
composition sequence generated by S that is not an escaping sequence.
To see (i) implies (iii), suppose that there are maps gn in S such that the composition
sequence Gn = g1 · · · gn generated by S is not an escaping sequence. Then, by Lemma 2.6,
there is a subsequence Gnk that converges to a Mobius transformation G. It follows that
G−1nk−1Gnk → I as k →∞. But
G−1nk−1Gnk = (g1 · · · gnk−1
)−1(g1 · · · gnk) = gnk−1+1 · · · gnk ,
so this sequence lies in S. It follows that I ∈ S, so S is not both semidiscrete and inverse
free.
Since each composition sequence generated by S is a subsequence of some composition
sequence generated by F , we see that (ii) implies (iii). On the other hand F ⊆ S, and so
(iii) implies (ii).
We say that a sequence of Mobius transformations is discrete if the set of transformations
that make up the sequence is a discrete subset of Aut(B3). Although escaping sequences
are all discrete, by definition, the converse does not hold; for example, the trivial sequence
I, I, I, . . . is discrete, but it is not an escaping sequence.
The preceding theorem gives a condition in terms of composition sequences for a semigroup
to be both semidiscrete and inverse free. The next, similar theorem gives a condition in
terms of composition sequences for a semigroup to be semidiscrete. The proof is similar,
so we only sketch the details.
Theorem 2.8. Let S = 〈F〉 be a semigroup. The following are equivalent:
(i) S is semidiscrete.
26 2. ON MOBIUS SEMIGROUPS
(ii) Every composition sequence generated by F is discrete.
(iii) Every composition sequence generated by S is discrete.
Proof. We first show that (iii) implies (i). Suppose that S is not semidiscrete. Then
there is a sequence of distinct transformations gn from S that converges to I. As before,
we can define a composition sequence Gn = g1 · · · gn for n = 1, 2, . . . , and, providing
that gn converges to I sufficiently quickly, we see that Gn converges to some Mobius
transformation G. Because the maps gn are distinct, it follows that Gn 6= G for infinitely
many n. Hence Gn is not discrete.
To see that (i) implies (iii), suppose that there are maps gn in S such that the composition
sequence Gn = g1 · · · gn generated by S is not discrete. Then we can choose a subsequence
Gnk consisting of distinct maps that converges to a Mobius transformation G. Therefore
G−1nk−1Gnk → I as k → ∞, where G−1nk−1
Gnk ∈ S \ I. This implies that S is not
semidiscrete.
Since each composition sequence generated by S is a subsequence of some composition
sequence generated by F , we see that (ii) implies (iii). Conversely since F ⊆ S it follows
that (iii) implies (ii).
5. Covering regions
In this section we introduce the concept of a covering region for a semigroup. These
appear to play a role for semidiscrete semigroups somewhat similar to the role played by
fundamental regions for discrete groups. A covering region for a semigroup S is a closed
subset D of B3 with nonempty interior such that
⋃g∈S∪I
g(D) = B3.
Trivially, B3 itself is a covering region for S. Let us denote the interior of a set X by int(X).
We say that a covering region D is a fundamental region for S if it satisfies the additional
property int(D) ∩ int(g(D)) = ∅ whenever g is a nonidentity element of S. This defini-
tion coincides with the usual definition of a fundamental region when S is a Kleinian group.
5. COVERING REGIONS 27
The next theorem due to Bell [10, Theorem 3] says that if a semigroup has a fundamental
region, then it is in fact a Kleinian group. Although we do not use the next theorem
again, we include it because [10] is difficult to obtain, and the proof presented there is
excessively complicated.
Theorem 2.9. Suppose that D is a fundamental region for a semigroup S such that
int(D) = D. Then S is a Kleinian group with fundamental region D.
Proof. Let p ∈ int(D) and choose any element g of S. As D is a fundamental region,
there is another element h of S ∪ I such that g−1(p) ∈ h(D). Hence p ∈ gh(D), and
since p ∈ int(D) and there are points in gh(int(D)) that accumulate at p (using that
int(D) = D) we see that int(D) ∩ gh(int(D)) 6= ∅. Therefore gh = I, and so h = g−1.
Since g was arbitrarily chosen, S is a group with fundamental region D. Hence it must be
a Kleinian group because it has a fundamental region.
We remark that in the theorem above, instead of assuming that int(D) = D, we can
suppose that S is countable. To see this we argue as follows: First note that D must
have non-empty interior, for otherwise D is nowhere dense (as D is closed), in which
case S(D) cannot cover B3 by the Baire category theorem. If g is an element of S then
g−1(int(D)) meets S(D) = S(∂D) ∪ S(int(D)). Note that ∂D is nowhere dense since D
is closed. Since S is countable the Baire category theorem tells us that S(∂D) has empty
interior. Hence S(∂D) does not cover g−1(int(D)), and so g−1(int(D)) ∩ h(int(D)) 6= ∅
for some h ∈ S. Hence int(D)∩gh(int(D)) 6= ∅ and it follows that gh = I, that is h = g−1.
We now consider covering regions that are not necessarily fundamental regions. In the
following lemma we refer to the inverse free part of a semigroup S, which is the set S\S−1.
Lemma 2.10. If D is a covering region for a semigroup S that is not a group, then D is
also a covering region for the inverse free part of S.
Proof. Choose any point p in B3 and let f ∈ S \ S−1. Then there is an element g of
S ∪ I such that f−1(p) ∈ g(D). Hence p ∈ fg(D). By Lemma 2.1, fg ∈ S \ S−1. As p
was chosen arbitrarily it follows that D is a covering region for S \ S−1.
28 2. ON MOBIUS SEMIGROUPS
We now give the main result of this section, which says that any semidiscrete semigroup
with a bounded covering region is in fact a group. The proof of this theorem is a good
example of how composition sequences themselves can be used as a tool for studying
semigroups.
Theorem 2.11. Any semidiscrete semigroup that has a bounded covering region is a
Kleinian group.
Proof. Let S be a semidiscrete semigroup and choose any point ζ in B3. As S has
a bounded covering region, we can choose a suitably large open hyperbolic ball D, with
(hyperbolic) radius r and centre ζ, such that⋃g∈S∪I
g(D) = B3.
As ∂D is compact, there is a finite subset T of S such that
∂D ⊆⋃g∈T
g(D).
We suppose T has been chosen with no redundancy, so that ρ(g(ζ), ζ) ≤ 2r for each g ∈ T .
We construct a composition sequence Gn = g1 · · · gn generated by T as follows.
(i) If n > 1 and ζ ∈ Gn−1(D), then choose gn arbitrarily from T .
(ii) Otherwise, choose gn from T such that ρ(Gn(D), ζ) is minimised.
Let ρn = ρ(Gn(D), ζ) and suppose n ≥ 2. If gn is chosen according to (i), then ρ(Gn−1(ζ), ζ) ≤
r, so
ρ(Gn(ζ), ζ) ≤ ρ(Gn(ζ), Gn−1(ζ)) + ρ(Gn−1(ζ), ζ) = ρ(gn(ζ), ζ) + ρ(Gn−1(ζ), ζ) ≤ 3r.
Hence ρn ≤ 3r. Suppose now that gn is chosen according to (ii). Note that the collection
Gn−1g(D) | g ∈ T covers Gn−1(∂D). Therefore
ρ(Gn−1(D), ζ) = ρ(Gn−1(∂D), ζ) ≥ ρ(Gn(D), ζ);
that is, ρn−1 ≥ ρn.
It follows that the sequence ρn is bounded, and so Gn is not an escaping sequence. The-
orem 2.7 now tells us that S is not inverse free. We know from Lemma 2.10 that the
inverse free part of S, if nonempty, also has D as a covering region. But applying the
5. COVERING REGIONS 29
above argument with S \S−1 replacing S gives a contradiction, because S \S−1 is inverse
free. Therefore S \ S−1 is empty, and so S is a semidiscrete group, that is, a Kleinian
group.
The converse is false: if S is a Kleinian group it may have no bounded covering region,
regardless of whether or not its limit set is the full ideal boundary.
Given a semidiscrete semigroup S and a point w in B3 that is not fixed by any element of
S \ I, we define the Dirichlet region for S centred at w to be the set
Dw(S) = z ∈ B3 | ρ(z, w) ≤ ρ(z, g(w)) for all g in S \ I.
This is the same definition of a Dirichlet region used in the theory of Kleinian groups.
Because the points S(w) do not accumulate at w or contain w, the Dirichlet region centred
at w always contains some open neighbourhood of w. We next verify that Dirichlet regions
of semidiscrete semigroups are indeed covering regions.
Theorem 2.12. Let S be a semidiscrete semigroup and let w be a point in B3 that is not
fixed by any element of S \ I. Then Dw(S) is a covering region for S.
Proof. The set Dw(S) is closed because it is an intersection of closed sets (similarly
it is also convex, although we do not use this fact). It has nonempty interior because, by
Theorem 2.3, the orbit of w under S does not accumulate at w. To verify the covering
property, suppose towards contradiction that there is a point z in B3 that does not lie in
h(Dw(S)) for any element h of S ∪ I. Then h−1(z) /∈ Dw(S), which implies that there
is a map g in S ∪ I that satisfies ρ(h−1(z), w) > ρ(h−1(z), g(w)). In other words,
(3) for all h ∈ S ∪ I there exists g ∈ S ∪ I such that ρ(z, h(w)) > ρ(z, hg(w)).
We use (3) to recursively define a composition sequence Fn = f1 · · · fn, where fi ∈ S∪I.
Let f1 = I. If f1, . . . , fn−1 have been defined, then, using (3), we let fn be an element of
S∪I that satisfies ρ(z, Fn−1(w)) > ρ(z, Fn−1fn(w)). The resulting composition sequence
Fn satisfies
ρ(z, w) > ρ(z, F1(w)) > ρ(z, F2(w)) > · · · .
30 2. ON MOBIUS SEMIGROUPS
As S is semidiscrete, Theorem 2.8 tells us that the composition sequence Fn is discrete. But
the sequence Fn(w) is bounded, so it must have a constant subsequence. This contradicts
the sequence of strict inequalities above. Hence, contrary to our earlier assumption,
z ∈⋃
h∈S∪I
h(Dw(S)),
and so Dw(S) is a covering region for S.
We remark that for a semidiscrete semigroup it is always possible to choose a point w ∈ B3
that is not fixed by any element of S \ I. The set of points in B3 that are fixed by some
element of S \ I are necessarily fixed points of elliptic transformations that lie in S.
Since S is semidiscrete and inverse free the elliptic transformations in S form a discrete
group, and in particular there are countably many of them. Hence the set of w ∈ B3 fixed
by an element of S \ I is contained in a countable union of geodesics of B3. It follows
that one can always define a Dirichlet region for semidiscrete semigroups.
Recall from Chapter 1 that the forward horospherical limit set of S, Λ+h (S), is the set of
points x in Λ+(S) such that, for any point ζ in B3, the orbit S(ζ) meets every horoball
based at x. The remainder of this section is concerned with proving that if the forward
horospherical limit set of a semidiscrete semigroup is the entire ideal boundary, then every
Dirichlet region of S is bounded. As we do this, it is convenient to work with the upper
half-space model of hyperbolic space, chiefly because in this model, a horoball based at∞
takes a convenient form. Recall that a point z ∈ H3 can be written in the form z = x+ tj
where x ∈ C and t > 0. We define the height of z to be ht[z] = t. Given x in C and w in
H3, let Hx(w) denote the open horoball that is based at x and tangent to w. So, working in
the upper half-space model, H∞(w) = z ∈ H3 | ht[z] > ht[w]. Let γz = z+ tj | t ≥ 0
denote the vertical geodesic from z to ∞. Then γz = γz ∪ ∞. Given distinct points u
and v in H3, we define
K(u, v) = z ∈ H3 | ρ(z, u) ≤ ρ(z, v).
Note that K(u, v) is the closure of K(u, v) in R3 (it is not the same as K(u, v)).
The next lemma is an elementary exercise in hyperbolic geometry.
5. COVERING REGIONS 31
Lemma 2.13. Suppose that u and v are distinct points in H3 and ht[u] < ht[v]. Then
K(u, v) ∩ γv = ∅.
We use the lemma to prove the following theorem, which we shall need in Chapter 3, and
helps us relate Dirichlet regions to horospherical limit sets.
Theorem 2.14. Let S be a semidiscrete semigroup, x ∈ C and w ∈ H3. Then x ∈ Dw(S)
if and only if the S-orbit of w does not meet Hx(w).
Proof. By conjugating, we can assume that x = ∞ and w = j. First suppose that
the S-orbit of j meets H∞(j). Then there is an element g of S such that g(j) ∈ H∞(j), and
so ht[g(j)] > 1. By Lemma 2.13, ∞ /∈ K(j, g(j)). Since Dj(S) is contained in K(j, g(j)),
we see that ∞ /∈ Dj(S).
Conversely, suppose that g(j) /∈ H∞(j) for every element g of S. Then for each map g, we
have γj ⊆ K(j, g(j)). Therefore γj ⊆ Dj(S), so ∞ ∈ Dj(S).
An immediate consequence of the theorem is the following important corollary.
Corollary 2.15. Let S be a semidiscrete semigroup. If Λ+h (S) = C, then S is a Kleinian
group and every Dirichlet region of S is bounded in H3.
Proof. Let Dw(S) be a Dirichlet region for S. We are given that Λ+h (S) = C, so the
orbit S(w) meets every horoball in H3. Theorem 2.14 now tells us that Dw(S) ∩ C = ∅.
Therefore Dw(S) is bounded in H3, and we infer from Theorem 2.11 that S is a Kleinian
group.
In the theory of Kleinian groups there are various theorems that relate the group to the
geometry of its Dirichlet region. For example, any Dirichlet region of a Fuchsian group has
finitely many sides (that is, maximal subsets of the region’s boundary that are contained
within geodesic segments) precisely when the group is finitely-generated. It is natural to
ask whether results of this type exist for semidiscrete semigroups. Corollary 2.15 is one
such result and in Theorem 2.17 we use the Dirichlet region to characterise exactly when
the forward limit set is equal to the full ideal boundary, at least for countable, semidiscrete
and inverse free semigroups.
32 2. ON MOBIUS SEMIGROUPS
For the rest of this chapter it is convenient to work in the ball model. For a semidiscrete
semigroup S, and any w ∈ B3 that is not fixed by any element of S, we define the ideal
boundary of Dw(S), which we denote by ew(S), to be Dw(S) ∩ S2, where the closure is
taken with respect to the Euclidean metric. As usual we abbreviate ew(S) to ew whenever
the semigroup is unambiguous. We note that ew is a closed subset of S2. We shall show
that, under certain assumptions, ew has a non-empty interior precisely when the forward
limit set is not the full ideal boundary. Towards this, we need an initial lemma, the proof
of which makes use of composition sequences in an essential way. We shall need to consider
the set of points in Λ+ that are the limit points of some composition sequence generated
by S. We denote the set of such points by Λ+q , and note that Λ+
q is forward invariant, and
so by Theorem 1.8 Λ+q is a dense subset of Λ+ whenever S is nonelementary.
Lemma 2.16. Suppose that S is a countable, semidiscrete and inverse free semigroup.
Then for all w ∈ B3 we have
S2 \ Λ+q ⊆ S(ew).
Proof. First note that since S is semidiscrete and inverse free, no point in B3 is fixed
by any element of S, and so ew, the ideal boundary of Dw(S), is defined. For a point
x ∈ S2 recall that Hx(w) is the horoball based at x such that w ∈ B3 lies on its boundary.
It suffices to show that for all x ∈ S2 \ Λ+q and all w ∈ B3, there exists g ∈ S such that
g−1(x) ∈ ew. Suppose towards contradiction that this is false, that is, there exists some
w ∈ B3 and x ∈ S2 \ Λ+q such that g−1(x) /∈ ew for all g ∈ S. Theorem 2.14 tells us that
x ∈ ew if and only if S(w) ∩Hx(w) = ∅. Hence
S(w) ∩Hg−1(x)(w) 6= ∅,
or equivalently
(4) gS(w) ∩Hx(g(w)) 6= ∅,
for all g ∈ S. Now choose any f1 ∈ S. Using (4) and setting g = f1, there exists f2 ∈ S
such that
f1f2(w) ∈ Hx(f1(w)).
5. COVERING REGIONS 33
Again appealing to (4) with g = f1f2, there exists f3 ∈ S such that
f1f2f3(w) ∈ Hx(f1f2(w)).
x
w
f1(w)
f1f2(w)
Hx(w)
Hx(f1(w))
Hx(f1f2(w))
Figure 2.2. The sequence Fn(w)
Continuing in this way (see Figure 2.2) we obtain a sequence fn in S such that
Fn+1(w) ∈ Hx(Fn(w)),
for each n ∈ N, where Fn = f1 · · · fn.
Since S is semidiscrete and inverse free, Theorem 2.7 tells us that the sequence Fn(w) ac-
cumulates only on the ideal boundary. Since each point in Fn(w) belongs to the horoball
Hx(f1(w)), the sequence of points Fn(w) converges to x. But this contradicts the assump-
tion that x /∈ Λ+q ; hence, for any w ∈ B3, we have x ∈ g(ew) for some g ∈ S. Since x was
chosen arbitrarily in the complement of Λ+q we have shown
S2 \ Λ+q ⊆ S(ew),
as required.
Theorem 2.11 says that if a semidiscrete semigroup has a bounded covering domain, then
it is a group. The next theorem is of a similar flavour.
Theorem 2.17. Suppose S is a countable, semidiscrete and inverse free semigroup. Then
for any w ∈ B3 the set ew has empty interior as a subspace of S2 if and only if Λ+ = S2.
34 2. ON MOBIUS SEMIGROUPS
Proof. Suppose ew contains an open set, U say. Since Λ+h is both dense in Λ+ and
disjoint from ew it follows that Λ+ does not meet U , and so Λ+ 6= S2.
For the converse, suppose that Λ+ 6= S2. Note that a closed set is nowhere dense if and
only if it does not contain an open set. Towards contradiction suppose that ew is nowhere
dense. Since S is countable S(ew) is a meagre set, that is, a countable union of nowhere
dense sets. It follows from the Baire category theorem that S(ew) has empty interior.
Now by Lemma 2.16 we have
int(S2 \ Λ+q ) ⊆ int(S(ew)) = ∅,
and so
S2 \ Λ+q = ∅,
where the interiors and closures are taken with respect to the Euclidean topology on S2.
But since Λ+q is dense in Λ+ it follows that Λ+ = S2, contrary to our assumption. Hence
ew cannot be nowhere dense after all.
It follows that for any countable, semidiscrete and inverse free semigroup S, if w and w′
are points in B3 that are not fixed by any element of S, then ew has empty interior if and
only if ew′ also has empty interior. In fact we have the following theorem, due to Edward
Crane.
Theorem 2.18. Suppose S is a countable, semidiscrete and inverse free semigroup, and
w and w′ are points in B3. Then ew is empty if and only if ew′ is also empty.
Proof. First suppose that ew is empty, and so Dw is a bounded covering region of
S. Since Dw is a compact fundamental region, it follows from Theorem 2.11 that S is a
Kleinian group. Let r be the diameter of the quotient orbifold M = B3/S, which is finite
because Dw is compact. Let π : B3 →M be the quotient map. Then for any point w′ ∈ B3
and any x ∈ B3 such that ρ(x,w′) > r, we can lift a shortest path in M from π(x) to π(w′)
to a path in B3 of the same length (at most r), starting at x. The lifted path finishes at
some g(w′) for g ∈ S \ I, and ρ(x,w′) > r ≥ ρ(x, g(w′)), so x /∈ Dw′(S). Hence Dw′(S)
is bounded, and ew′ = ∅ as required.
6. CONSTRUCTING NEW SEMIDISCRETE SEMIGROUPS FROM OLD 35
6. Constructing new semidiscrete semigroups from old
We now describe a technique based on one of the well known combination theorems of
Klein, given in the introduction of [30] and which we now state.
Theorem 2.19. Suppose G1 and G2 are two finitely-generated Kleinian groups with fun-
damental regions D1 and D2 respectively. Further suppose that the interior of D1 contains
the boundary and exterior of D2, and the interior of D2 contains the boundary and exterior
of D1. Then the group G generated by G1 ∪G2 is discrete and D1 ∩D2 is a fundamental
region for G.
Given distinct points u and v in B3, recall that K(u, v) = z ∈ B3 : ρ(z, u) ≤ ρ(z, v).
Notice that g(K(u, v)) = K(g(u), g(v)) for any Mobius transformation g. If S is a semidis-
crete semigroup, and w is a point of B3 that is not fixed by any element of S, then we can
express the Dirichlet region for S centred at w as
Dw(S) =⋂
g∈S\I
K(w, g(w)).
Then
B3 \Dw(S) =⋃
g∈S\I
K(g(w), w),
where K(g(w), w) = z ∈ B3 | ρ(z, g(w)) < ρ(z, w) is the interior of K(g(w), w).
Theorem 2.20. Suppose that S1 and S2 are semidiscrete semigroups and w is a point in
B3 that is not fixed by any nontrivial element of S1 or S2. Suppose that B3 \Dw(S−12 ) ⊆
Dw(S1) and B3 \ Dw(S−11 ) ⊆ Dw(S2). Then the semigroup T generated by S1 ∪ S2 is
semidiscrete. Furthermore, if S1 and S2 are inverse free, then so is T .
Proof. Observe that, for g ∈ S1 \ I and h ∈ S2 \ I, we have
K(g(w), w) ⊆ B3 \Dw(S1) ⊆ Dw(S−12 ) ⊆ K(w, h−1(w)),
which implies that K(g(w), w) ⊆ K(w, h−1(w)). Hence h(K(g(w), w)) ⊆ K(h(w), w),
and similarly g(K(h(w), w)) ⊆ K(g(w), w). It follows that h maps B3 \ Dw(S1) into
B3 \Dw(S2), and g maps B3 \Dw(S2) into B3 \Dw(S1).
36 2. ON MOBIUS SEMIGROUPS
Next, observe that g(w) ∈ B3 \ Dw(S1) and h(w) ∈ B3 \ Dw(S2). Choose f ∈ T \ I.
By writing f as a reduced word with letters from S1 and S2, we see that f(w) belongs to
either B3 \Dw(S1) or B3 \Dw(S2). However, S1 and S2 are semidiscrete, so w is contained
in the interior of both Dw(S1) and Dw(S2). It follows that T is semidiscrete.
Furthermore, if S1 and S2 are inverse free, then the same argument shows that no nontrivial
reduced word with letters from S1 and S2 is equal to the identity element. Hence T is
inverse free also.
Theorem 2.17 says that if S is countable, semidiscrete and inverse free, then ew(S) has
nonempty interior if and only if Λ+(S) 6= S2. This implies that if Λ+(S) 6= S2 and
Λ−(S) 6= S2, then by Theorem 2.20 we can always choose another Mobius transformation
f that does not lie in S, such that the semigroup T = 〈f ∪ S〉 is semidiscrete and in-
verse free. The construction also shows that Λ+(T ) is contained in the Euclidean closure
of (B3 \Dw(S)) ∪K(f(w), w), and so we can choose f such that Λ+(T ) 6= S2.
Theorem 2.20 can be used as a tool for modifying known semidiscrete semigroups to gener-
ate new semidiscrete semigroups, as the following example illustrates. Let S1 = 〈f0, f1, f2〉
and S2 = 〈g0, g1, g2〉, where each generator is defined by the parameters in the table fol-
lowing Figure 2.5. Both semigroups are semidiscrete and inverse free. The Dirichlet
regions of S1, S−11 , S2 and S−12 with respect to the point w = 0 are shown in the fig-
ures below. The Dirichlet region of S1 (the complement of the red region) and S−11 (the
complement of the blue region) are shown in Figure 2.3. The Dirichlet region of S2 (the
complement of the red region) and S−12 (the complement of the blue region) are shown
in Figure 2.4. Notice that the red region in Figure 2.3 does not meet the blue region in
Figure 2.4, although it does meet the red region in Figure 2.4. Similarly the red region in
Figure 2.4 does not meet the blue region in Figure 2.3, although it does meet the red region
in Figure 2.3. Hence, by the procedure described above, the semigroup S generated by
S1∪S2 is also semidiscrete and inverse free. The Dirichlet regions of S (the complement of
the red region) and of S−1 (the complement of the blue region), are depicted in Figure 2.5.
6. CONSTRUCTING NEW SEMIDISCRETE SEMIGROUPS FROM OLD 37
f0
f1
f2
Figure 2.3. The generators of S1 together with the Dirichlet regions of
S1 and S−11
g0
g1
g2
Figure 2.4. The generators of S2 together with the Dirichlet regions of
S2 and S−12
38 2. ON MOBIUS SEMIGROUPS
g0
g1
g2
f0
f1
f2
Figure 2.5. The generators of S together with the Dirichlet regions of S
and S−1
Transformation h αh βh tr2(h)
f0 eπi e0πi 22.05
f1 e0.25πi e1.25πi 22.05
f2 e1.75πi e0.75πi 22.05
g0 e0.5πi e1.5πi 202.005
g1 e0.35πi e0.65πi 10.453
g2 e1.35πi e1.65πi 10.453
CHAPTER 3
Semigroups that fix the unit circle
In this chapter we begin to present the main results of this thesis. Although many of the
auxiliary results hold in all dimensions, most of the main results within this section ap-
ply only to subsemigroups of Aut(D). Our arguments usually do not generalise to higher
dimensions because Theorem 3.8, upon which other results rely, depends on the topolog-
ical property that any connected subset of the ideal boundary consisting of at least two
points contains a non-empty open subset. Nevertheless, some of our results do extend to
subsemigroups of Mobius transformations that fix the unit circle. This means that we can
include Mobius transformations that transpose the two connected components of C \ S1,
as well as those that fix both components.
The group of Mobius transformations that fix the unit circle, which we denote Aut(S1), is
significant in the theory of continued fractions because, thinking in the upper half-plane
model, although z 7→ z + 1 fixes the upper half-plane, the map z 7→ 1/z transposes the
upper and lower half-planes. The unit circle in the complex plane identifies with the equa-
tor in the unit ball model. This means that, in the unit ball model, since any element of
Aut(S1) fixes the equator it must also fix the Euclidean disc
(x, y, z) ∈ B3 | z = 0
con-
tained in B3 whose boundary is the equator. Indeed, Aut(S1) is exactly the set of isome-
tries (both orientation-preserving and orientation-reversing) of
(x, y, z) ∈ B3 | z = 0
endowed with the hyperbolic metric inherited from B3. Each element of Aut(S1) either
fixes the southern hemisphere (which corresponds to the unit disc in the complex plane)
or transposes the northern and southern hemispheres. In this chapter we are concerned
with subsemigroups of Aut(D), however, by working in the ball model and considering
the action of Aut(S1) on
(x, y, z) ∈ B3 | z = 0
, our arguments that do not require this
action to be orientation-preserving (for example Theorem 3.1) also hold for subsemigroups
39
40 3. SEMIGROUPS THAT FIX THE UNIT CIRCLE
of Aut(S1).
We now survey the topics discussed within this chapter. In Section 2 we develop an
understanding of composition sequences generated by finitely-generated semidiscrete sub-
semigroups of Aut(D). In particular we prove the following theorem.
Theorem 3.1. Let S = 〈F〉 be a finitely-generated subsemigroup of Aut(D). The following
statements are equivalent:
(i) every composition sequence generated by S is an escaping sequence;
(ii) every composition sequence generated by S converges ideally;
(iii) S is semidiscrete and inverse free.
(iv) every composition sequence generated by F is an escaping sequence;
(v) every composition sequence generated by F converges ideally;
This tells us that for any semidiscrete subsemigroup of Aut(D), every composition se-
quence converges ideally if and only if the group part is empty. We also describe how
composition sequences can behave when S has a nonempty group part.
In Section 3 we give an algorithm for deciding whether or not a two-generator subsemi-
group of Aut(D) is semidiscrete, semidiscrete and inverse free, or neither. Our methods
also give a classification of semidiscrete two-generator semigroups (Theorem 3.20). Our
strategy is similar to the geometric approach for classifying discrete two-generator groups
initiated by Matelski [31] and others, and completed by Gilman [17]. However, our task
is much simpler than that of classifying discrete two-generator groups. This is because
by far the most difficult case in the groups classification arises when the two generators
are loxodromic maps with intersecting axes. In contrast, the semigroup generated by two
such maps is easily seen to be semidiscrete.
It is well known that any group of Mobius transformations is either elementary, Fuchsian,
or dense in Aut(D). Section 4 is concerned with proving a counterpart of this result for
subsemigroups of Aut(D). It has a similar statement, but with an additional category. Let
J be a nontrivial closed interval of S1. Throughout this chapter whenever we say ‘closed
3. SEMIGROUPS THAT FIX THE UNIT CIRCLE 41
interval’ we shall mean a nontrivial closed interval (that is nonempty, and not a single
point or the whole unit circle) contained in the unit circle. We define Mob(J) to be the
collection of Mobius transformations in Aut(D) that map J within itself. Clearly Mob(J)
is a semigroup, which is not semidiscrete. We shall prove the following result.
Theorem 3.2. Let S be a subsemigroup of Aut(D). Then S is either
(i) elementary,
(ii) semidiscrete,
(iii) contained in Mob(J), for some nontrivial closed interval J , or
(iv) dense in Aut(D).
If S is finitely-generated, then most semigroups that are contained in Mob(J) for some
interval J (of type (iii)) are also semidiscrete (of type (ii)). There are some exceptions to
this, however; for example, (now using the upper half-plane model) the semigroup con-
tained in Aut(H2) and generated by the maps z 7→√
2z, z 7→ 12z and z 7→ z + 1. This
semigroup is contained in Mob([0,+∞]), but it is not elementary, semidiscrete or dense in
Aut(H2). In Section 5 we will classify the small collection of finitely-generated semigroups
that are not of types (i), (ii) or (iv). We will see that S does not lie in this special collection
of semigroups if no two members of a generating set for S are loxodromic with the same
attracting and repelling fixed points. This version of Theorem 3.2 for finitely-generated
semigroups represents a significant generalisation of a result of Barany, Beardon and Carne
[2, Theorem 3], who proved that any semigroup generated by two non-commuting trans-
formations, one of which is elliptic of infinite order (that is, an elliptic map g for which
gn 6= I for any n ∈ N), is dense in Aut(D).
In Section 5 we prove another theorem which has a familiar counterpart in the theory of
Fuchsian groups. This counterpart theorem says that a nonelementary group of Mobius
transformations is discrete if and only if each two-generator subgroup of the group is
discrete. Our theorem only applies to finitely-generated subsemigroups of Aut(D). Unfor-
tunately, there is also a bothersome class of semigroups that we must treat as exceptional
cases, which we now describe.
42 3. SEMIGROUPS THAT FIX THE UNIT CIRCLE
Definition Let S be a semigroup that lies in Mob(J), for some closed interval J . Suppose
that the collection of elements of S that fix J as a set forms a nontrivial discrete group.
Suppose also that one of the other members of S (outside this discrete group) fixes one
of the end points of J . In these circumstances we say that S is an exceptional semigroup;
otherwise it is a nonexceptional semigroup.
These terms should be treated in a similar way to how we treat the terms elementary and
nonelementary in the theory of groups or semigroups; that is, exceptional semigroups, like
elementary groups or semigroups, are easy to handle, but do not obey all the laws satisfied
by the more typical cases. We emphasise that it is easy to tell whether a finitely-generated
semigroup is exceptional by examining its generating set, as we explain later. The main
result of Section 5 is the following.
Theorem 3.3. Suppose S is a finitely-generated nonexceptional semigroup S contained in
Aut(D), that is not elementary of infinite type. Then S is semidiscrete if and only if every
two-generator semigroup contained in S is semidiscrete.
We use the results on exceptional semigroups developed in this section to give a version of
Theorem 3.2 for finitely-generated semigroups. This is Corollary 3.30 and can be regarded
as a (coarse) classification of finitely-generated subsemigroups of Aut(D).
Section 6 covers our final theorem, on limit sets of semigroups. There appears to be a
relationship between the size of the intersection Λ+(S) ∩Λ−(S) and the size of the group
S ∩ S−1. For example, if Λ+(S) ∩Λ−(S) is finite, then the group S ∩ S−1 is either empty
or elementary (because if it is not elementary, then its limit set, which is contained in
both Λ+(S) and Λ−(S), is perfect). The main result of this section is about when the
intersection Λ+(S) ∩ Λ−(S) is large.
Theorem 3.4. Let S be a finitely-generated nonelementary subsemigroup of Aut(D). Then
Λ+(S) = Λ−(S) if and only if S is a group.
In fact, we prove that if Λ−(S) ⊆ Λ+(S), then S is a group.
1. A CLOSER LOOK AT ELEMENTARY SUBSEMIGROUPS OF Aut(H2) 43
Before proceeding, we take a more detailed look at elementary subsemigroups of Aut(D),
or equivalently elementary subsemigroups of Aut(H2), and in particular give the promised
version of Theorem 1.7 from Chapter 1 which has been specialised to semigroups contained
in Aut(H2).
1. A closer look at elementary subsemigroups of Aut(H2)
Theorem 3.5. Let S be an elementary semigroup of finite type contained in Aut(H2).
Then S is conjugate in Aut(H2) to a semigroup that is contained in one of the following
sets, depending on |Λ−| and |Λ+|:
(i) |Λ−| = |Λ+| = 0: z 7→ (az − b)/(bz + a) | a2 + b2 = 1;
(ii) |Λ−| = |Λ+| = 1: z 7→ z + a | a ∈ R or z 7→ λz | λ > 0;
(iii) |Λ−| = |Λ+| = 2: z 7→ λz, z 7→ −λ/z | λ > 0;
(iv) (a) |Λ−| = 1, |Λ+| =∞: z 7→ az + b | a ≤ 1, b ∈ R;
(b) |Λ−| =∞, |Λ+| = 1: z 7→ az + b | a ≥ 1, b ∈ R.
The proof of this theorem is elementary, and similar to the proof of the classification of ele-
mentary semigroups of Mobius transformations given in [15, Theorem 2.11]. Accordingly,
we only sketch a justification of the theorem.
Proof of Theorem 3.5. Suppose that one of Λ− and Λ+ is empty. Then S contains
only elliptic transformations, which each fix a common point. If, by conjugation, we assume
that the fixed point is i, then S is contained in z 7→ (az − b)/(bz + a) | a2 + b2 = 1.
Suppose now that |Λ+| = 2. After conjugating, we can assume that Λ+ = 0,∞. Since
Λ+ is forward invariant under S, every element of S must either fix 0 and∞, or interchange
them. The elements of Aut(D) that fix both 0 and ∞ have the form z 7→ λz, λ > 0, and
the elements of Aut(D) that interchange 0 and ∞ have the form z 7→ −λ/z, λ > 0. The
former type are loxodromic (unless λ = 1) and the latter type are elliptic. Since ∞ ∈ Λ+,
it must be an attracting fixed point of some map z 7→ λz, λ > 1. Hence 0 ∈ Λ−, and
similarly ∞ ∈ Λ−. Therefore Λ− = 0,∞.
We are left to consider the cases when one of |Λ−| or |Λ+| has order 1. If both have order 1,
then we are in case (ii); we omit the details. If only one has order 1, say |Λ−| = 1 (the
other case is similar), then by conjugation we can assume that Λ− = ∞. The elements
44 3. SEMIGROUPS THAT FIX THE UNIT CIRCLE
of Aut(D) that fix ∞ have the form z 7→ az + b, where a > 0 and b ∈ R. If a > 1, then
this map is loxodromic and has a repelling fixed point at b/(1− a). As all repelling fixed
points of loxodromic elements of S belong to Λ−, we see that S is of type (iv)(a).
Later on we will require a more precise understanding of those finitely-generated semi-
groups of type (ii). The following lemma satisfies our needs: it classifies those semigroups
that are contained in the group z 7→ z + a | a ∈ R. This task is the same as classifying
additive semigroups of real numbers, and so some form of this lemma almost certainly
features in the literature already.
Lemma 3.6. Let S be a semigroup generated by maps z 7→ z + ai, i = 1, . . . , n. Then
exactly one of the following statements is true.
(a) Either ai ≥ 0 for i = 1, . . . , n or ai ≤ 0 for i = 1, . . . , n, in which case S is semidiscrete.
(b) For every pair of indices i and j, we have ai/aj ∈ Q, provided aj 6= 0, and one of these
quotients ai/aj is negative. In this case, S is a discrete group.
(c) Otherwise, there are two maps z 7→ z+ ai and z 7→ z+ aj that generate a dense subset
of z 7→ z + a | a ∈ R.
Proof. In case (a), it is clear that S is semidiscrete.
In case (b), choose any of the numbers ai (other than 0), and let aj be of opposite sign
to ai. Then mai = −naj for some coprime positive integers m and n. Choose positive
integers u and v such that vn−um = ±1. Let α = ai/n = −aj/m. Then vai +uaj = ±α.
It follows that S contains one of the maps z 7→ z + α or z 7→ z − α. But S also contains
z 7→ z + nα and z 7→ z −mα. So S must contain both z 7→ z + α and z 7→ z − α, and in
particular it must contain z 7→ z − ai. Therefore S is a group, and one can use a short,
standard argument from the theory of discrete groups to prove that S is discrete.
In case (b), we can choose a pair of nonzero numbers ai and aj such that ai/aj /∈ Q. We
can assume that ai and aj have opposite signs: if at first they do not, then replace one of
them by another number ak of the opposite sign (if you are careful about which of ai and
aj you replace, then you can be sure that the quotient of the two numbers you end up
with is irrational). Now let un/vn be a sequence of rational numbers, where un, vn > 0,
1. A CLOSER LOOK AT ELEMENTARY SUBSEMIGROUPS OF Aut(H2) 45
that satisfies ∣∣∣∣aiaj +unvn
∣∣∣∣ < 1
2v2n.
Then |vnai + unaj | → 0 as n→∞, and we know that vnai + unaj 6= 0. From this we see
that the maps z 7→ z+ vnai +unaj accumulate at the identity, so 〈z 7→ z+ ai, z 7→ z+ aj〉
is not semidiscrete.
The group z 7→ z + a | a ∈ R considered in the lemma is the group of orientation-
preserving Euclidean isometries of R. The other group in Theorem 3.5(ii) is z 7→ λz | λ >
0, which is the group of orientation-preserving isometries of one-dimensional hyperbolic
space I, which in this case is modelled by the positive imaginary axis. The metric spaces
R and I are isometric using the transformation x 7→ ex. It follows that the classification of
finitely-generated semigroups of z 7→ λz | λ > 0 is much the same as the classification of
Lemma 3.6; if S is generated by maps z 7→ λiz, i = 1, . . . , n, then we can define ai = log λi
and cases (a)–(c) remain valid, with minor adjustments.
We have now examined class (ii) of Theorem 3.5 in detail. Let us briefly survey the
other classes in turn, without proofs. Throughout our analysis we assume that S is a
finitely-generated semigroup that is contained in a set G, where G is specified for each
class.
Class (i). Let G = z 7→ (az − b)/(bz + a) | a2 + b2 = 1. Then S is either dense in G or
else it is a finite cyclic subgroup of G.
Class (iii). Let G = z 7→ λz, z 7→ −λ/z | λ > 0. Assume that S contains a map of the
form z 7→ −λ/z (otherwise S has been dealt with already in class (ii)). Then S is a group,
so either it is dense in G or else it is conjugate in Aut(B3) to a discrete group generated
by z 7→ −1/z and z 7→ µz, where µ > 1.
Class (iv)(a). Let G = z 7→ az + b | a ≤ 1, b ∈ R. Define S1 = f(z) = z + b | f ∈ S.
Then S1 is finitely-generated, so either it is empty, or else it is one of the types (a)–(c)
determined in the first part of the analysis of class (ii), above. If S1 is of type (c), then S
is not semidiscrete, but it is not dense in G. If S1 is empty or of types (a) or (b), then S
is a semidiscrete semigroup.
Class (iv)(b). This is similar to the previous case (switch between S and S−1 to move
from one class to the other).
46 3. SEMIGROUPS THAT FIX THE UNIT CIRCLE
2. Proof of Theorem 3.1
In this section it is convenient to use the upper half-plane model of the hyperbolic plane,
whose ideal boundary is the extended real line R = R∪∞. Before proving Theorem 3.1,
we need a basic lemma.
Lemma 3.7. Suppose that S is a semigroup generated by a finite set F of Mobius trans-
formations, and suppose that Fn is an escaping composition sequence generated by F . Let
L = Λ+c (Fn). Then for any positive integer m, F−1m (L) ⊆ Λ+
c (S).
Proof. Let p ∈ L. Choose ζ ∈ H2. Then there is a sequence of positive integers
n1, n2, . . . such that the sequence Fni(ζ) converges conically to p. Hence the sequence
F−1m Fni(ζ) converges conically to F−1m (p). Now, if ni > m, then F−1m Fni ∈ S, and so
F−1m (p) ∈ Λ+c (S), as required.
The lemma remains true if we replace the forward conical limit sets with forward limit
sets; however, we shall not need this alternative statement.
Theorem 3.8. Suppose that S is a semigroup generated by a finite set F contained in
Aut(D). Suppose also that there is an escaping composition sequence Fn generated by F
that does not converge ideally. Then Λ+c (S) = S1, unless Λ−(S) = q for some point q,
in which case Λ+c (S) is either equal to S1 or to S1 \ q.
Proof. Working in the upper half-plane model, choose any point ζ in H2. Let k =
maxρ(ζ, f(ζ)) | f ∈ F. We are given that the sequence Fn(ζ) accumulates, in the
chordal metric, at two distinct points a and b in R. Suppose there is a point d other than
a and b that is not a forward conical limit point of Fn. By conjugating Fn if need be we
can assume that d =∞ and a < b. Let γc be the hyperbolic geodesic with one end point
c inside (a, b) and the other at ∞. Let Γc = z ∈ H2 | ρ(z, γc) < k. Notice that
ρ(Fn−1(ζ), Fn(ζ)) = ρ(ζ, fn(ζ)) ≤ k.
It follows that infinitely many terms from the sequence Fn(ζ) lie in Γc. This infinite set
inside Γc cannot accumulate at ∞ because ∞ is not a forward conical limit point of Fn,
so it must accumulate at c. Hence c is a forward conical limit point of Fn.
2. PROOF OF THEOREM 3.1 47
We have shown that (a, b) ⊆ Λ+c (Fn). Choose a point y in (a, b). Since Λ−c (F−1n ) =
Λ+c (Fn), we can apply Lemma 1.3 to deduce the existence of a sequence of positive integers
n1, n2, . . . and two distinct points p and q in R such that F−1ni (y) → p and F−1ni (z) → q
for z 6= y. It follows that any point u in R \ q is contained in F−1ni (a, b), providing ni is
sufficiently large. Lemma 3.7 tells us that F−1ni (a, b) ⊆ Λ+c (S), so we see that R \ q ⊆
Λ+c (S).
Finally, suppose that Λ−(S) 6= q. If there exists an element f of S that does not fix q,
then q = f(v) for some point v in R \ v. As Λ+c (S) is forward invariant under S, we
see that it is the whole of R. Otherwise if every element of S fixes q, then S contains a
loxodromic element with attracting fixed point q, in which case q ∈ Λ+c (S) and so again
we have q ∈ Λ+c (S) = R.
The exceptional case in which Λ−(S) = q and Λ+c (S) = R \ q certainly can arise.
For example, let S =⟨z 7→ 1
2z, z 7→ z + 1, z 7→ z − 1⟩. Clearly Λ−(S) = ∞, as all the
generators of S fix ∞, and ∞ is a repelling fixed point of z 7→ 12z. One can easily
construct a composition sequence generated by S that is an escaping sequence that does
not converge ideally, and one can check that Λ+c (S) contains R. However, ∞ /∈ Λ+
c (S)
because all elements of S have the form z 7→ az + b, where a ≤ 1 (and so the S-orbit of a
point in H2 cannot accumulate conically to ∞).
A Kleinian group G is said to be cocompact if its quotient space B3/G is compact, or
equivalently, if G has a compact fundamental region. We can now prove the main result
of this section, from which we will deduce Theorem 3.1.
Theorem 3.9. Let S = 〈F〉 be a finitely-generated semidiscrete subsemigroup of Aut(D)
such that |Λ−(S)| 6= 1. Then the following statements are equivalent:
(i) every composition sequence generated by F that is an escaping sequence converges
ideally;
(ii) S is not a cocompact Fuchsian group.
Proof. Suppose first that S is a cocompact Fuchsian group. Then it has a finite-sided
compact fundamental polygon D and the images of D under S tessellate D. Choose z0 in
the interior of D and suppose f1(D), f2(D), . . . , fk(D) are the neighbouring tiles of D. Let
48 3. SEMIGROUPS THAT FIX THE UNIT CIRCLE
D0 = D,D1, D2, . . . be a sequence of tiles such that Di is adjacent to Di+1 for each i. Let
zi be the point of S(z0) that lies in Di. We can choose the sequence Di so that the sequence
zi accumulates at two points of the unit circle S1, but does not accumulate in D. For each
n ≥ 0 there exists a unique element gn ∈ S such that gn(D) = Dn. The neighbours of
Dn are gn(fi(D)), and one of these is Dn+1 = gn+1(D), and so gn+1 = gnfin+1 for some
in+1 ∈ 1, . . . , k. Therefore gn = fi1 · · · fin defines a composition sequence generated by
the finite set F = f1, . . . , fk. Since gn(z0) = zn, the sequence is escaping but does not
converge ideally.
Suppose, conversely, that there is a composition sequence generated by F that is an
escaping sequence but does not converge ideally. By Theorem 3.8, we see that Λ+c (S) = S1.
Hence Λ+h = S1 and so Corollary 2.15 then tells us that S is a group with a compact
Dirichlet region: it is a cocompact Fuchsian group.
Suppose F is a finite subset of Aut(D) that generates a semidiscrete semigroup S such that
|Λ−(S)| 6= 1. Further suppose that S is not a cocompact Fuchsian group. If Fn = f1 · · · fnis a composition sequence generated by F , then there are two cases: either Fn escapes,
or not. If Fn escapes, then it converges ideally by Theorem 3.9. Otherwise Fn does not
escape, and for all large enough n each fn belongs to the group part of S. To see this,
we have by Lemma 2.6 that along some subsequence nk, the sequence Fnk converges to
a Mobius transformation. Hence, as in the proof of Theorem 2.7, F−1nkFnk+1
converges to
the identity. Since S is semidiscrete, this means that F−1nkFnk+1
= fnk+1 · · · fnk+1is equal
to the identity for all large enough k. It follows that for all large enough n, each fn lies in
the group part of S.
Theorem 3.9 fails if we remove the hypothesis |Λ−(S)| 6= 1; we have seen an example of
this already, just before the statement of the theorem. Let us now use this theorem to
prove Theorem 3.1.
Proof of Theorem 3.1. The equivalence of (i) and (iii) was established already in
Theorem 2.7. That (ii) implies (i) is immediate. The most substantial part of Theorem 3.1
is the implication of (ii) from (iii).
3. TWO-GENERATOR SEMIGROUPS 49
Suppose then that S is an inverse free semidiscrete semigroup, generated by a finite set
F . We must prove that every composition sequence generated by S converges ideally. By
the equivalence of (i) and (iii), we know that every composition sequence generated by
S is an escaping sequence. If we regard any composition sequence generated by S as a
subsequence Fnk of a composition sequence Fn generated by F , then Theorem 3.9 tells us
that, provided |Λ−(S)| 6= 1, Fn converges ideally, and so Fnk also converges ideally. In the
special case where |Λ−(S)| = 1 we can assume, by conjugating, that Λ−(S) = ∞. Now
by examining Class (iv)(a) in Theorem 3.5, we see that there is a real number a, such
that all elements of S either map the interval (−∞, a) within itself, or map the interval
(a,+∞) within itself. Either way Theorem 3.8 tells us that every composition sequence
generated by S that is an escaping sequence must converge ideally.
We see that (i) and (iv) are equivalent by Theorem 2.7. To show that (v) and (ii) are
equivalent we argue just as in the proof of Theorem 2.7: any composition sequence gen-
erated by S is a subsequence of some composition sequence generated by F , and so (v)
implies (ii). Conversely since F ⊆ S it follows that (ii) implies (v).
3. Two-generator semigroups
In this section we classify the semidiscrete two-generator subsemigroups of Aut(D), and we
determine which of them are Fuchsian groups and which are inverse free semigroups. This
section can be thought of as a version of Gilman and Maskit’s paper [18] for semidiscrete
semigroups. There, the authors give an algorithm that determines whether or not a two-
generator subgroup of Aut(D) is discrete, which omits the more complicated case of two
loxodromic generators with intersecting axes, which Gilman deals with in [17]. This section
relates only to subsemigroups of Aut(D), and so throughout all Mobius transformations
are understood to be of this type. The task of classification is simpler if we handle
elementary two-generator semigroups separately to the main classification of two-generator
semigroups.
Theorem 3.10. The semigroup S = 〈g, h〉 generated by two nontrivial Mobius transfor-
mations g and h in Aut(D) is elementary if and only if
(i) g and h have a common fixed point in D; or
50 3. SEMIGROUPS THAT FIX THE UNIT CIRCLE
(ii) g and h are both elliptic of order 2; or
(iii) one of g and h is loxodromic and the other is elliptic of order 2, and the fixed point
of the elliptic generator lies on the axis of the loxodromic generator.
Proof. To begin note that if f ∈ Aut(D) fixes w ∈ C then it also fixes 1/w. We will
work though the cases listed in Theorem 1.5.
If S fixes a point x in H3 then S consists of elliptic transformations. The axis of any
elliptic transformation in Aut(D) has ideal endpoints w and 1/w for some w in D. So S
consists of elliptic transformations that fix the unique point w in D such that the geodesic
that ends at w and 1/w passes though x, and we are in case (i).
Suppose S is elementary but does not fix any point in H3. Then either S fixes a single
point w in C or a pair of points v, w in C. In the first case, w must lie on the unit circle,
since 1/w is also fixed by S, so again we are in case (i). In the second case, we cannot have
v = 1/w because then v would be in D and w /∈ D or vice versa, and S would fix pointwise
the geodesic joining v and w in H3. We must therefore have v = 1/v and w = 1/w, so v
and w are on the unit circle. Either g and h both fix v and w, so we are in case (i), or they
both exchange v and w, so we are in case (ii), or one fixes them and the other exchanges
them, so we are in case (iii).
There are examples of semigroups of both finite and infinite type in case (i), however
every semigroup in case (ii) and case (iii) are of finite type. Using Theorem 3.10 it is
straight forward to decide which two-generator elementary semigroups of finite type are
semidiscrete and which are inverse free. The only two-generator elementary semigroups
that are of infinite type are described in the following proposition, and are all semidiscrete
and inverse free.
Theorem 3.11. Suppose g and h are two nontrivial Mobius transformations in Aut(D)
that generate an elementary semigroup of infinite type. Then S is semidiscrete and inverse
free.
Proof. Suppose g and h are nontrivial Mobius transformations in Aut(D) and that
S = 〈g, h〉 is an elementary semigroup of infinite type. By the comments following Theorem
1.5, both g and h fix some point w in C. If w is not on the unit circle, then S fixes a
3. TWO-GENERATOR SEMIGROUPS 51
point in the unit disc (either w or 1/w), and so both limit sets are empty, contrary to our
assumption that S is of infinite type. Hence w lies on the unit circle and both generators
are not elliptic transformations. If at least one of g or h is a parabolic transformation,
then it is easy to see that at least one of Λ+(S) or Λ−(S) is equal to w. Hence both g
and h are loxodromic maps. If w = αg = αh or w = βg = βh, then again, one of the limit
sets of S is equal to w. If the set of fixed points of g is equal to the set of fixed points
of h, then both Λ+(S) and Λ−(S) are contained in this set of fixed points. Since S is of
infinite type, it follows that either S or S−1 is conjugate in Aut(B3) to the configuration
shown in Figure 3.1.
g h
Figure 3.1. The axes of g and h
It now suffices to show that if the generators of S are configured as in Figure 3.1, then S
is semidiscrete and inverse free. Choose a point z in the component of S1 \ βg, αh that
does not contain the common fixed point αg = βh. Let J be the closed interval bounded
by z and the common fixed point αg = βh that contains αh. Both g and h map J strictly
within itself, and so by Theorem 2.5, S is semidiscrete and inverse free.
Now we have classified all two-generator semigroups in the cases where the generators have
fixed points in common. We now proceed to consider two-generator semigroups for which
any point fixed by one generator is not fixed by the other. We first consider the case where
at least one generator is elliptic. Clearly S cannot be both semidiscrete and inverse free.
However, if exactly one generator is elliptic and of finite order, and the composition of the
two generators is not elliptic, then we have an example of a two-generator semidiscrete
semigroup whose group part and inverse free part are both nonempty (see case (b) in
52 3. SEMIGROUPS THAT FIX THE UNIT CIRCLE
Theorem 3.12). In fact, examples of this type are the only two-generator semidiscrete
semigroups whose group part and inverse free part are both nonempty. Throughout the
rest of this section it is convenient to denote the group part of S, that is S ∩ S−1, by G.
Theorem 3.12. Suppose S = 〈g, h〉 where g is not elliptic and h is elliptic. Then exactly
one of the following is true:
(a) h has infinite order and S is not semidiscrete;
(b) h has finite order, ghk is not elliptic for any k = 0, . . . , order(h)−1, and S is semidis-
crete;
(c) h has finite order, and for some k ∈ 0, . . . , order(h)− 1 the element ghk is an elliptic
transformation of infinite order, and S is not semidiscrete;
(d) h has finite order, and for some k ∈ 0, . . . , order(h)− 1 the element ghk is an elliptic
transformation of finite order, and S is the group generated by g and h.
Moreover any two-generator semidiscrete semigroup whose group part and inverse free part
are both nonempty must arise from case (b).
Proof. Suppose h is elliptic and g is either loxodromic or parabolic. If h has infinite
order, then S is not semidiscrete and we have case (a). Now suppose h is of finite order,
and by conjugating if necessary, we can suppose h has fixed point 0. Let m be the order
of h. We can replace h with hd, where d and m are coprime, without affecting 〈g, h〉.
Therefore we can assume that the angle of rotation of h is 2π/m.
If g is loxodromic, choose `γ to be the unique geodesic that is perpendicular to the axis
of g and passes though 0. Let `α be the unique geodesic that is perpendicular to the axis
of g and such that g = αγ, where α and γ denote reflection in `α and `γ respectively.
If g is parabolic, choose `γ to be the unique geodesic that passes though 0 and lands at
the fixed point of g. Let `α be the unique geodesic that lands at the fixed point of g and
such that g = αγ, where again α and γ denote reflection in `α and `γ respectively.
Now let us choose β to be the (unique) reflection such that h = γβ. Let `β denote the line
of reflection of β, and note that the angle between `β and `γ is π/m. We define `βk to be
3. TWO-GENERATOR SEMIGROUPS 53
the geodesic line passing through 0 obtained by rotating `γ by an angle of kπ/m, and let
βk denote reflection in `βk . Note that β0 = γ, β1 = β and hk = βkβ0.
There are now two possibilities, either `α cuts `βk in D for some k = 0, . . . ,m − 1, or
otherwise. In the latter case, for some k the line `α is contained in a sector of angle π/m
subtended between `βk and `βk+1. Let Γ = 〈βk, βk+1, α〉 and let q denote whichever of k,
k + 1 is even. Since β0 = h−q/2βqhq/2 and h = βk+1βk ∈ Γ, it follows that g = αβ0 must
lie in Γ. Hence we have S ⊆ Γ. By Poincare’s Polygon Theorem 3.15, the shaded region
bounded by `βk , `βk+1and `α shown in Figure 3.2 is a fundamental region for Γ. Since Γ
is discrete S is semidiscrete, and so we have case (b).
`α
`βk `βk+1
πm
Figure 3.2. Fundamental region for Γ
Now let Y denote the closed arc contained in S1 whose end points are end points of `βk
and `βk+1, and which contains the end points of `α. Let X = Y ∪ h(Y ) ∪ · · · ∪ hm−1(Y ).
Then X is a nontrivial closed subset of S1 that satisfies h(X) = X and, since g = αβ0,
satisfies g(X) ⊆ Y . It follows that 〈g, h〉 is not equal to the group generated as a group
by g and h.
Earlier in the proof we assumed that `α did not meet `βk in D for any k = 0, . . . ,m − 1.
Now suppose that `α meets `βk in D. It follows that ghk = αβk is elliptic. If ghk has
infinite order, then S is not semidiscrete and we have case (c). Otherwise we are left
with the case where h and ghk are elliptic elements of finite order. Since g = ghkhm−k,
it follows that S is equal to⟨h, ghk
⟩. This means that S is generated by two finite order
54 3. SEMIGROUPS THAT FIX THE UNIT CIRCLE
elliptic transformations, and so S is a group. This group is equal to the group generated
by g and h, and we have case (d).
Finally, we suppose S = 〈g, h〉 is an arbitrary semidiscrete semigroup with nonempty
group and inverse free parts. By Lemma 2.2, we know that the group part is generated by
just one of the transformations g or h – let us say h. Therefore h must be elliptic of finite
order. As 〈g, h〉 is semidiscrete and has group part 〈h〉, the element g cannot be elliptic;
it must be loxodromic or parabolic. Furthermore, the map ghk cannot be elliptic for any
k, otherwise⟨h, ghk
⟩is a group and is equal to S. We are now left with the possibility
described in case (b).
We emphasise that in case (d) of the theorem above, S is a group and so S is semidiscrete if
and only if the group generated by the elliptic transformations g and ghk (for appropriate
k) is discrete. Whether or not this is the case can be determined by the Gilman-Maskit
algorithm detailed in [18].
We now consider two-generator semigroups where both generators are not elliptic and the
generators’ fixed points do not coincide. It is useful to describe a pair of transforma-
tions g, h as antiparallel if neither is elliptic, the set of fixed points of g is disjoint from
the set of fixed points of h, and no proper closed interval of the unit circle whose end
points are fixed points of g or h is mapped strictly within itself by either g or h. Intu-
itively this means that g and h ‘point’ in opposite directions. Figure 3.3 shows the three
possible configurations of the transformations (up to conjugation in Aut(D) and replace-
ment of S with S−1). The horoballs in the last two diagrams represent parabolic elements
which fix the horoballs setwise, and move points on the horoball in the direction indicated.
The following theorem handles the remaining cases where g and h are not antiparallel.
Theorem 3.13. If g and h are both not elliptic, are not antiparallel, and S = 〈g, h〉 is
nonelementary, then S is semidiscrete and inverse free.
3. TWO-GENERATOR SEMIGROUPS 55
hg hg hg
Figure 3.3. Possible configurations of antiparallel transformations g and h
Proof. By first replacing S by S−1 if necessary, the generators of S are conjugate to
one of the possibilities shown in Figure 3.4.
hg
hg
hg hg
Figure 3.4. Possible configurations of generators in Theorem 3.13
In each case it is easy to identify a closed interval contained in the unit circle that is
mapped strictly inside itself by both generators, and so S is semidiscrete and inverse free
by Theorem 2.5.
It remains for us to classify two-generator semigroups that are nonelementary and have
antiparallel generators. We give two theorems that can be used as the basis for an al-
gorithm that can determine whether or not a semigroup generated by two antiparallel
transformations g and h is semidiscrete, semidiscrete and inverse free, or neither. In prin-
ciple, given g and h, an upper bound for the runtime of the algorithm can be computed
before execution. In fact we show that when g and h are antiparallel, S is semidiscrete and
inverse free if and only S contains no elliptic maps, and in that case the group generated
by g and h is itself discrete and elliptic-free. The following lemma records an observation
that we shall use repeatedly.
Lemma 3.14. Suppose S = 〈g, h〉 where neither of g, h is an elliptic element of finite order.
If S contains an elliptic element of finite order, then S is equal to the group generated by
g and h.
56 3. SEMIGROUPS THAT FIX THE UNIT CIRCLE
Proof. Suppose f ∈ S is elliptic of order n = 2, 3, . . .. Written as a word in g and h,
f is not a power of g nor a power of h, and so the word contains both the letters g and h.
Since fn = I, we have a word in both letters g and h that equals the identity. Without
loss of generality we can suppose that f takes the form f = gf1, where f1 is a nonempty
word. Then (gf1)n = I and so g−1 = f1(gf1)
n−1 ∈ S. Since the letter h features in the
word f and g−1 ∈ S, we can similarly deduce that h−1 ∈ S. It follows that S = G.
As a consequence of the above, whenever we encounter an elliptic element in S we can
check if S is semidiscrete as follows. Either the elliptic element is of infinite order, and
so S is not semidiscrete, or the elliptic element has finite order, in which case S = G by
Lemma 3.14 and S is semidiscrete exactly when G is discrete. In the latter case we can
appeal to the algorithm given in [18] to decide if G is discrete or not.
Our next theorem deals with the more straightforward case where at least one of g or
h is parabolic. In order to prove it we shall make use of Poincare’s Polygon Theorem
[3, Theorem 9.8.4], which we now describe.
Let P be a convex polygon, that is a hyperbolically convex subset of D whose boundary
can be written as the finite union of non-trivial geodesic segments and connected subsets
of S1. The geodesic segments may land on the ideal boundary at both ends, one end, or
neither end. A side of P is a maximal subset of the boundary of P that is either contained
within a geodesic segment, or a connected subset of S1. A vertex of P is any end point of
one of the sides of P , which may lie in S1 or in D. A vertex that lies on S1 is called an ideal
vertex. A side-pairing transformation of P is a hyperbolic isometry gs (not necessarily
orientation preserving), that maps one side s of P bijectively to another side t such that
gs(P ) ∩ P = t.
Theorem 3.15. Poincare’s Polygon Theorem. Let P be a polygon in D equipped with a
set of side-pairing transformations, gs, one for each side of P , such that gt = g−1s if gs
pairs s with t. Suppose there exists ε > 0 with the following properties.
(i) For each vertex x0 ∈ D there are vertices x1, . . . , xn of P and hyperbolic isometries
f0, . . . , fn+1 such that f0 = fn+1 = I and for each j = 0, . . . , n there is a side s
such that fj+1 = fjgs. Further suppose that if Nj = z ∈ P | ρ(xj , z) < ε then
3. TWO-GENERATOR SEMIGROUPS 57
the sets fj(Nj) do not overlap and have union B(x0, ε).
(ii) For each ideal vertex y0 of P there are ideal vertices y1, . . . , yn of P and hyperbolic
isometries f0, . . . , fn+1 such that f0 = fn+1 = I and for each j = 0, . . . , n there
is a side s of P such that fj+1 = fjgs, and fj(N′j) are contiguous and do not
overlap, where N ′j = z ∈ P | |yj − z| < ε.
Then the group G generated by the side-pairing transformations gs is discrete, P is a
fundamental region for G, and each relation of G arises from a vertex in D and finite
composition of its associated maps f0, . . . , fn+1 that fixes P equated with the identity.
Notice that if all the side-pairing transformations for polygon P are orientation preserv-
ing, and all the vertices of P are ideal, then the group generated by the side-pairing
transformations has no non-trivial relations: it is a free group.
We shall make use of the following lemma in both the subsequent theorem and in Theo-
rem 3.19
Lemma 3.16. Suppose g1 = g and h1 = h are Mobius transformations, and nj is a sequence
of positive integers. Further suppose we have finite sequences g1, . . . , gk and h1, . . . , hk that
satisfygj+1, hj+1
=gnjj hj , g
nj+1j hj
for j = 1, . . . , k− 1. If the group generated by gk
and hk is free, then the semigroup 〈g, h〉 does not contain the identity.
Proof. Let Gj be the group generated by gj and hj . Clearly Gj+1 ⊆ Gj . Since
gj = (gnj+1j hj)(g
njj hj)
−1
and
hj = g−njj (g
njj hj),
we see that Gj ⊆ Gj+1. Hence Gj = G1 for each j. If a group is free, then it is freely
generated by any choice of generators. It follows that since Gk is free, so is G1, and in
particular, the semigroup generated by g and h does not contain the identity.
We remark that the conclusion of the lemma holds if the assumptiongj+1, hj+1
=
gnjj hj , g
nj+1j hj
is replaced with
gj+1, hj+1
=hjg
njj , hjg
nj+1j
.
58 3. SEMIGROUPS THAT FIX THE UNIT CIRCLE
Theorem 3.17. Suppose g and h are antiparallel Mobius transformations, and h is par-
abolic. Then S is semidiscrete and inverse free if and only if ghn is not elliptic for any
n ∈ N. Moreover, exactly one of the following is true.
(a) The map ghn is not elliptic for all n ∈ N, in which case S is free and the group
generated by g and h is discrete and elliptic-free;
(b) for some n ∈ N the map ghn is an elliptic transformation of infinite order and S is
not semidiscrete; or
(c) for some n ∈ N the map ghn is an elliptic transformation of finite order, and S is a
group that is semidiscrete precisely when S is a discrete group.
Proof. We first consider the case where g is parabolic. Let ` be the geodesic landing
at the fixed points of g and h, and denote the reflection in ` by σ. We can choose a
geodesic `g such that g = σgσ where σg is the reflection in `g, and geodesic `h such that
h = σσh where σh is the reflection in `h. Since g and h are antiparallel, ` does not separate
`h from `g in D. Since h and g are parabolic, `g and `h land at the fixed points of g and h
respectively. We have the two cases shown in Figure 3.5: either `g meets `h in D, or not.
If `g and `h do meet in the unit disc, then gh = σgσh fixes the point of intersection, and
`
`g`h
`
`h`g
Figure 3.5. The two cases of Theorem 3.17 when both g and h are parabolic
so gh is an elliptic map. If gh is elliptic of infinite order, then S is not semidiscrete and
we have case (b). Otherwise gh has finite order, in which case Lemma 3.14 tells us that
S is a group. Hence S is semidiscrete exactly when S is a discrete group, and we fall into
case (c).
Otherwise the geodesics `, `h and `g bound a region U in D, and we show that we have
the situation describe by case (a). By Poincare’s Polygon Theorem 3.15, if Γ is the group
of conformal and anticonformal Mobius transformations generated by the reflections σ, σg
3. TWO-GENERATOR SEMIGROUPS 59
and σh, then Γ is discrete and contains the group generated by S. It follows that S is
semidiscrete. We note that U ∪ σ(U) is a fundamental polygon for the group generated
by S, which we denote by T , and each vertex lies on the ideal boundary. It follows from
Poincare’s theorem that T , and so S itself, does not contain any elliptic elements. Since
there are no nontrivial vertex relations, Poincare’s theorem further shows that T is free as
a group generated by g and h, and hence S is free as a semigroup generated by g and h.
It follows that ghn is not elliptic for any n ∈ N. Indeed, S is a Schottky semigroup, and so
does not contain the identity. To see this, let Jg be the closed interval on the unit circle
whose end points are those of `g and that does not contain the fixed point of h. Similarly
let Jh be the closed interval on the unit circle whose end points are those of `h and that
does not contain the fixed point of g, as shown in Figure 3.6.
`
`h`g
UJg
σ(Jh)
Figure 3.6. The domain U with intervals Jg and σ(Jh)
Both g and h map Jg ∪ σ(Jh) strictly within itself, and so S is a Schottky semigroup, as
claimed.
We now consider the case where g is loxodromic and h is parabolic. Staying in the unit
disc model, we conjugate so that i is the fixed point of h and the imaginary axis is per-
pendicular to the axis of g. We let σ denote the reflection in the imaginary axis, and
choose σg such that g = σgσ. For each n = 0, 1, 2, . . . we let σn be the reflection such that
hn = σσn. Note that σ = σ0. We let `, `g and `n denote the lines of reflection of σ, σg and
σn respectively. We discriminate between two cases: either σg and σn meet in D for some
n = 0, 1, 2, . . ., or not. If they do meet, then ghn = σgσn is an elliptic transformation,
60 3. SEMIGROUPS THAT FIX THE UNIT CIRCLE
and, just as in the case where both g and h were parabolic, we have case (b) if ghn has
infinite order and case (c) if ghn has finite order.
Otherwise σg and σn do not meet in D for each n = 0, 1, 2, . . .. It follows that for some
n the lines `n, `g and `n+1 bound a region U , as shown in Figure 3.7. We show that S is
described by case (a). By Poincare’s Polygon Theorem 3.15, the group Γ = 〈σn, σn+1, σg〉
`n
`n+1
`
`g
U
Figure 3.7. The lines `, `n, `n+1 and `g, and the region U
is discrete, with fundamental region U . Moreover since all the vertices of U are ideal,
the only relations on Γ are the trivial ones: σ2n = σ2n+1 = σ2g = I. Let T be the group
generated by ghn+1 and ghn. Certainly T ⊆ Γ, and since T is orientation preserving it
does not contain σn, σn+1 or σg. It follows that T has no nontrivial relations, in other
words, T is a free group. It now follows from Lemma 3.16 and the remarks following
its proof that S does not contain the identity. To see that S is semidiscrete, note that
σk−1 = σkσk+1σk for all k ∈ N, hence by induction both σ0 and σ1 lie in Γ. It follows
that both the generators g and h also lie in Γ, and so S is semidiscrete since Γ is discrete.
Finally, since the group generated by g and h is orientation preserving and contained in
Γ, it must be free, and in particular, contains no elliptic transformations.
In the case where h is parabolic and g loxodromic, we remark that it is not necessary
to test whether or not ghn is an elliptic transformations for every n ∈ N, in order to
determine which of the cases (a), (b), or (c) our semigroup 〈g, h〉 falls into. Indeed, for
a particular g and h, the proof gives the basis of an algorithm that identifies the largest
such n we must test. If both g and h are parabolic, then n = 1 suffices; we only need to
3. TWO-GENERATOR SEMIGROUPS 61
verify whether or not gh is an elliptic map.
Our next task is to treat the case where g and h are both loxodromic and antiparallel. We
first require a result on the geometry of quadrilaterals, which can be found in [3, Theorem
7.17.1].
Lemma 3.18. Consider the quadrilateral with side lengths b1, a2, a1 and b2, listed in the
anticlockwise sense. Let φ denote the angle between b1 and b2, and suppose all the other
internal angles are right angles.
a1
a2
b1
b2
φ
Then we have:
(i) sinh a1 sinh a2 = cosφ, and
(ii) cosh a1 = cosh b1 sinφ.
Theorem 3.19. Suppose g, h are two antiparallel loxodromic maps. Then S = 〈g, h〉 is
semidiscrete and inverse free if and only if S is elliptic-free. Moreover, whenever S is
semidiscrete and inverse free:
(i) the group generated by g and h is discrete and elliptic-free; and
(ii) we can find two elements in S that generate G, and G is a free group.
If S does contain an elliptic element, then either S is not semidiscrete, or S = G and G
is discrete.
Proof. We work in the disc model. By applying a conjugation if necessary, we can
assume that the axes of g and h are symmetrical about both the real and imaginary axes,
as is the case in both examples of Figure 3.8. Let ` be the geodesic contained in the
imaginary axis that cuts the axes of both g and h orthogonally, and let σ be the reflection
in `. There exists a unique geodesic `g orthogonal to the axis of g such that g = σgσ
(where σg is the reflection in `g) and a unique geodesic `h orthogonal to the axis of h such
62 3. SEMIGROUPS THAT FIX THE UNIT CIRCLE
that h = σσh (where σh is the reflection in `h). Since elliptic maps are characterised by
the existence of fixed points inside D, it follows gh = σgσh is elliptic exactly when `g and
`h meet in D.
For a loxodromic Mobius transformation f , let ρ(f) denote half its translation length and
let ax(f) denote the axis of f . Hence ρ(g) and ρ(h) are the (hyperbolic) distances from `
to `g and from ` to `h, respectively. Without loss of generality we suppose ρ(h) ≥ ρ(g).
There are two cases to consider, which are shown in Figure 3.8: either `h meets ax(g) in
D, or otherwise.
g
h
`
`h
g
h
`
`h
Figure 3.8. The cases where g, h is decidable and otherwise
Suspending for now our assumption that ρ(h) ≥ ρ(g), we say that any pair of antiparal-
lel transformations g and h is decidable if, after conjugating so that their axes enjoy the
symmetry shown in Figure 3.8, either `h does not meet ax(g) in D (they may meet on
the ideal boundary) or `g does not meet ax(h) in D. We use the term ‘decidable’ because
whenever g, h has this property, it is relatively easy to determine if S is semidiscrete and
inverse free, using a procedure that we now describe. In fact we show that when g, h
is decidable and ρ(h) ≥ ρ(g), then S is semidiscrete and inverse free if and only if gnh is
not elliptic for any n ∈ N. To prove this, we use similar ideas to those used in the proof
of Proposition 3.17. For each n = 0, 1, 2, . . . consider the geodesic `n orthogonal to ax(g)
such that gn = σnσ. Note that `0 = `, `1 = `g and that the (hyperbolic) distance from
` to `n is nρ(g). Since g, h is decidable and ρ(h) ≥ ρ(g), then `h does not meet ax(g).
If for some n ∈ N the two geodesics `n and `h meet in D, then gnh = σnσh is elliptic.
Otherwise for some least n = 0, 1, 2, . . ., the lines `, `n+1 and `h bound a region in D. We
3. TWO-GENERATOR SEMIGROUPS 63
``n
`n+1
`h
U
g
h
Figure 3.9. The lines `, `n, `n+1 and `h, and the region U , for the decid-
able case
now apply Poincare’s theorem to the region U bounded by `n, `n+1 and `h (see Figure 3.9),
and infer that the group Γ of conformal and anticonformal maps generated by σn, σn+1
and σh is discrete. As the orientation-preserving subgroup of Γ, G is discrete and so S is
semidiscrete. The region U ∪σn(U) is a fundamental region for G, and by considering the
action of gnh and gn+1h on U ∪σn(U), it follows from Poincare’s theorem that G is freely
generated as a group by gnh and gn+1h, and moreover contains no elliptic elements. (The
semigroup may contain parabolic elements, for example if either of `n or `n+1 meet `h on
the ideal boundary.) It now follows from Lemma 3.16 that S = 〈g, h〉 does not contain the
identity. The preceding discussion can be used as the basis for an algorithm that deter-
mines whether or not a decidable pair of antiparallel Mobius transformations generates a
semidiscrete and inverse free semigroup. More importantly the above helps to establish a
more general algorithm that applies when the transformations are not decidable, which is
the case we consider next.
64 3. SEMIGROUPS THAT FIX THE UNIT CIRCLE
For antiparallel Mobius transformations g and h we define the following quantity
d(g, h) = sinh[max(ρ(h), ρ(g))]. sinh[ρ(ax(g), ax(h))].
By considering the shaded quadrilateral in Figure 3.10, it follows from an application of
Lemma 3.18 that `h does not meet ax(g) in D precisely when
sinh[ρ(h)]. sinh[ρ(ax(g), ax(h))] ≥ 1.
Hence g, h is decidable precisely when d(g, h) ≥ 1.
g
h
`h
`
Figure 3.10. The case where d(g, h) = 1
In the case where g and h are not decidable, we proceed as follows: We let (g0, h0) = (g, h)
and construct a finite sequence (g0, h0), (g1, h1), . . . , (gm, hm) for some m ∈ N. For each
k = 0, . . . ,m the maps gk and hk are loxodromic and antiparallel elements of S, and
generate G as a group. Suppose (gk, hk) has been constructed. We show that either:
(i) gnkhk is elliptic for some positive integer n; or
(ii) at least one of gk and hk is parabolic; or
(iii) d(gk, hk) ≥ 1; or
(iv) we can find antiparallel gk+1, hk+1 ∈ S that generate G as a group, and satisfy
d(gk+1, hk+1) > d(gk, hk)/(1− d(gk, hk)2).
If at any stage in the construction we encounter case (i), then S is not semidiscrete and
inverse free. In case (ii) we can appeal to Proposition 3.17 and conclude that either S
contains an elliptic map, or the group generated by gk and hk, which by construction is
G itself, is discrete and freely generated by gk and hk. In the latter case, by invoking
3. TWO-GENERATOR SEMIGROUPS 65
Lemma 3.16, we see that I /∈ S and therefore S is semidiscrete and inverse free. Simi-
larly in case (iii) we have d(gk, hk) ≥ 1, so that g, h is decidable. This means we can
use the procedure described above to either conclude that S contains an elliptic element,
or that G is discrete and freely generated by gk and hk. Hence S is semidiscrete and,
by Lemma 3.16, is also inverse free. In case (iv) we can continue the sequence and find
antiparallel loxodromic maps gk+1, hk+1 ∈ S that generate G as a group. The sequence
must terminate because if dk is a sequence such that 0 < dk < 1 and dk+1 > dk/(1− d2k),
then dk cannot remain in (0, 1) for all k. Moreover if m is the least positive integer such
that dm is not in (0, 1), then dm ≥ 1. This means that we cannot continue constructing
the sequence (gk, hk) indefinitely without meeting one of the cases (i), (ii) or (iii).
If at any point of the algorithm we encounter an elliptic element, it is of finite or infinite
order. If the elliptic element has infinite order then S is not semidiscrete. Otherwise the
elliptic element has finite order, and so S = G by Lemma 3.14; and so S is semidiscrete
if and only if G is discrete. A finite time algorithm that determines whether or not G
is discrete is given in [18]. Conversely if G is discrete then every elliptic element in G
has finite order, and hence the same is true of S. This proves the last sentence in the
statement of the theorem.
It remains to show that (gk, hk) can be constructed as claimed. For clarity of exposition
we take k = 0 so that g = g0 and h = h0. As in the decidable case, we let `n be the
geodesic orthogonal to ax(g) such that gn = σσn, and so gnh = σhσn. If `n meets `h for
some n ∈ N, then gnh is elliptic and we have case (i). Another possibility is that `h meets
either of `n or `n+1 on the ideal boundary. In this case gnh and gn+1h can be seen to
be antiparallel, and are either both parabolic, or one is parabolic and one loxodromic. In
these cases we can appeal to Proposition 3.17 and infer that either S contains an elliptic
map, or G is discrete and elliptic-free. Otherwise for some n, `h separates `n from `n+1,
and gnh and gn+1h are both loxodromic (see Figure 3.11). We let (g1, h1) = (gnh, gn+1h)
and set n1 equal to n. Since gnh = σnσh, its axis is orthogonal to both `n and `h. Sim-
ilarly gn+1h = σn+1σh so that its axis is orthogonal to `n+1 and `h. It can be seen from
66 3. SEMIGROUPS THAT FIX THE UNIT CIRCLE
the geometry that gnh and gn+1h are antiparallel. To verify this, observe that ax(gnh) is
the mutual perpendicular of `n and `h, and, since gnh = σnσh, we have gnh parallel with
h. Similarly gn+1h is parallel with g. Since the axes of gnh and gn+1h are both orthog-
onal to `h, if they were to meet in D then they must be equal setwise, so we must have
σh(`n) = `n+1. But this cannot be, for then ax(g), as the mutual perpendicular of `n and
`n+1, must be perpendicular to `h. (This is more clear upon considering Figure 3.11 and
conjugating so that `h is a Euclidean diameter.) Since g = (hgn)−1(hgn+1), the element
g belongs to the group generated by gnh and gn+1h, and so h also belongs to this group.
Hence gnh and gn+1h generate G as a group.
g
ha b
c
p
q rz
x
y
vu
`h
`n
`n+1
gnh
gn+1h
ψφ
Figure 3.11. The geometry of constructing (g1, h1) from (g, h)
For points α and β in D we let [α, β] denote the geodesic line joining α to β, and we ab-
breviate ρ(α, β) to ραβ. Notice that [u, v] is contained in ax(gn+1h) and [y, x] is contained
3. TWO-GENERATOR SEMIGROUPS 67
in ax(gnh). With g1 = gnh and h1 = gn+1h we want to find
d(g1, h1) = sinh[max(ρ(gn+1h), ρ(gnh))]. sinh[ρ(ax(gnh), ax(gn+1h))].
We have ρ(ax(gnh), ax(gn+1h)) = ρcy+ρcb+ρbv and ρ(gn+1h) = ρxy ≤ max(ρ(gn+1h), ρ(gnh)).
Hence to prove d(g1, h1) > d(g0, h0)/(1− d(g0, h0)2) it suffices to show that
(5) sinh[ρcy + ρcb + ρbv] sinh[ρxy] > d0/(1− d20).
In order to proceed we consider the geometry of several hyperbolic quadrilaterals found
within the figure above and repeatedly apply Lemma 3.18. Considering the quadrilateral
czyx we have
(6) cosh ρcy sinψ = cosh ρzx
and
(7) sinh ρzx sinh ρxy = cosψ.
Considering the quadrilateral pqcb we have
(8) cosh ρ = cosh ρcb sinψ,
where ρ = ρ(ax(g), ax(h)). Finally, considering the quadrilateral abuv we obtain
(9) cosφ = sinh ρuv sinh ρbv.
Equations (6) and (7) yield
(10) cosh ρcy =1
sinψ
√cos2 ψ
sinh2 ρxy+ 1
and so
(11) sinh ρcy =1
tanψ tanh ρxy.
We can now express sinh(ρcy + ρcb + ρbv) in terms of sinh or cosh of ρcy, ρcb and ρbv, and
eliminate these terms using equations (11), (8) and (9) above.
68 3. SEMIGROUPS THAT FIX THE UNIT CIRCLE
We have
sinh(ρcy + ρcb + ρbv) = sinh ρcy(cosh ρcb cosh ρbv + sinh ρcb sinh ρbv) +
cosh ρcy(sinh ρcb cosh ρbv + sinh ρbv cosh ρcb)
≥ sinh ρcy cosh ρcb cosh ρbv
=1
tanψ tanh ρxy
cosh ρ
sinψ
√cos2 φ
sinh2 ρuv+ 1
≥ cosh ρxy cosh ρ cosψ
sin2 ψ sinh ρxy,
and so
sinh(ρcy + ρcb + ρbv) sinh ρxy >d0
(1− d20)cosh ρxy cosh ρ
>d0
(1− d20)
as required.
It follows from Theorem 3.19 that if g and h are antiparallel loxodromic Mobius trans-
formations, then S = 〈g, h〉 is semidiscrete if and only if S is contained in a discrete
group. This is not true when g and h are not antiparallel. Some of the ideas used in the
proof above are taken from [18]; however, the proof given there uses the fact that if the
group generated by g and h is discrete and nonelementary, then Jørgensen’s inequality
(see [3, Theorem 5.4.1]) is satisfied. That is,
|tr2(g)− 4|+ |tr(ghg−1h−1)− 2| ≥ 1.
We could not assume Jørgensen’s inequality, since it was not clear beforehand that if S is
semidiscrete and generated by two antiparallel loxodromic Mobius transformations, then
S is contained in a discrete group. Neither was it clear beforehand how to determine
whether or not S contains an elliptic map; indeed, if S does contain an elliptic map, then
the proof of Theorem 3.19 can be used to find one such map explicitly.
Theorem 3.19 also has relevance to the study of groups, for it serves as a test to determine
whether or not the group generated by two loxodromic transformations with disjoint axes
3. TWO-GENERATOR SEMIGROUPS 69
is discrete and elliptic-free, without making use of Jørgensen’s inequality.
We now collate results obtained in this section into the following theorem.
Theorem 3.20. Suppose the semigroup S = 〈g, h〉 is nonelementary. Then S is semidis-
crete if and only if:
(i) exactly one of g and h is elliptic (h say), h has finite order, and ghk is not elliptic for
each k ∈ 0, . . . , order(h)− 1; or
(ii) S is a Fuchsian group; or
(iii) g and h are not elliptic and not antiparallel; or
(iv) g and h are antiparallel, and S is elliptic-free.
These cases are mutually exclusive. The results in this section and their proofs collec-
tively constitute a basis for an algorithm that decides whether or not g and h generate a
semidiscrete semigroup S. Moreover, it can be determined into which of these four cases
S belongs. Theorem 3.20 also classifies those nonelementary two-generator semigroups
that are semidiscrete and inverse free – these are exactly the cases (iii) and (iv). Case (i)
describes a rather small class of semigroups, and is dealt with in Theorem 3.12.
Proof. Since S is nonelementary the set of fixed points of g is disjoint from the set
of fixed points of h. Suppose now that at least one of g and h is elliptic. If both g and
h are elliptic and one has infinite order, then S is not semidiscrete. Otherwise both have
finite order and S = G. Hence S is semidiscrete precisely when G is discrete, which can
be determined by the Gilman-Maskit algorithm given in [18]. If exactly one generator is
elliptic then we examine cases (a)–(d) in Theorem 3.12. In cases (a) and (c), S is not
semidiscrete. Case (b) exactly describes case (i) above. In case (d), both g and h are
elliptic elements of finite order and S is a group. Either S is a Fuchsian group, which is
case (ii), or otherwise S is not discrete, and hence S is not semidiscrete.
It remains to consider the cases where both generators are either parabolic or loxodromic
and have disjoint sets of fixed points. We consider separately the cases where g and h
70 3. SEMIGROUPS THAT FIX THE UNIT CIRCLE
are antiparallel, and otherwise. If g and h are not antiparallel then we have the situa-
tion described in Theorem 3.13. Otherwise g and h are antiparallel, and the procedure
described in the proof of Theorem 3.17 (when at least one of g or h is parabolic), or Theo-
rem 3.19 (when g and h are both loxodromic), can decide whether or not S is semidiscrete,
semidiscrete and inverse free, or neither. If S is not semidiscrete and inverse free then
the procedure identifies an elliptic transformation in S. If this elliptic transformation is
of infinite order then S is not semidiscrete. Otherwise S = G and we must again defer to
the Gilman-Maskit algorithm detailed in [18] in order to determine whether or not G is
discrete.
4. Proof of Theorem 3.2
In this section we prove Theorem 3.2 as well as other results that we shall need later. Our
first theorem has a well known counterpart in the theory of Fuchsian groups. Recall that
αf and βf denote the attracting and repelling fixed points (respectively) of a loxodromic
Mobius transformation f .
Theorem 3.21. Suppose that S is a semigroup of Mobius transformations that includes
loxodromic elements and is not elementary of infinite type. Then for every pair of open
subsets U and V of S2 such that U meets Λ+(S) and V meets Λ−(S), there is a loxodromic
element of S with attracting fixed point in U and repelling fixed point in V .
Proof. First suppose that S is elementary. Then by Proposition 1.7, S is one of the
types (c), (d), (e) or (f), as stated in the proposition. By inspection and using Theorem
1.8 (i), it can be seen that each of these elementary semigroups satisfies the conclusion of
this theorem.
Now suppose that S is not elementary. By Theorem 1.8, there are loxodromic maps f
and g in S such that αf ∈ U and βg ∈ V , and since Λ+(S) is perfect we can assume
that αf 6= βg. If either αg ∈ U or βf ∈ V , then we have found a loxodromic map of the
required type. Suppose instead that αg /∈ U and βf /∈ V . For the moment, let us assume
also that βf 6= αg (so that no two of αf , βf , αg and βg are equal). We can choose pairwise
disjoint open intervals Af , Bf , Ag and Bg such that αf ∈ Af , βf ∈ Bf , αg ∈ Ag, βg ∈ Bg,
and moreover Af ( U and Bg ( V . Now let n be a sufficiently large positive integer
4. PROOF OF THEOREM 3.2 71
such that fn maps the complement of Bf into Af , and f−n maps the complement of Af
into Bf . Suppose also that n is large enough such that gn maps the complement of Bg
into Ag, and g−n maps the complement of Ag into Bg. One can now check that the map
h = fngn satisfies h(Af ) ( Af and h−1(Bg) ( Bg. Hence h is a loxodromic element of S
that satisfies αh ∈ U and βh ∈ V , as required.
It remains to consider the case where βf = αg (and, as before, αf , βf and βg are pairwise
distinct). Since S is nonelementary there is an element s of S that maps βf to a point
outside the set αf , βf , βg. Let kn = sgn. Then kn is loxodromic for sufficiently large
values of n. Moreover, αkn → s(αg) and βkn → βg as n→∞. Choose n such that αf , βf ,
αkn and βkn are pairwise distinct, and βkn ∈ V . We can now apply the argument of the
preceding paragraph with kn replacing to g to obtain a loxodromic element of S with the
desired properties.
The assumption that S includes loxodromic elements and is not elementary of infinite type,
is equivalent to assuming that S is not an elementary semigroup of type (a), (b) or (g),
as described in Proposition 1.7. In particular Theorem 3.21 holds for all nonelementary
semigroups.
A Gromov hyperbolic metric space is a metric space within which we can draw triangles
and these triangles are always ‘thin’. More precisely, a Gromov hyperbolic metric space
enjoys the following properties. Firstly, every two points in the space are the end points
of some minimizing geodesic. This means that for each set of three distinct points in the
space we can associate a ‘triangle’ whose vertices are the three points, and whose three
edges are the minimizing geodesics between each pair of vertices. Finally, there exists
δ > 0 such that for each triangle every point on one of its sides lies a distance at most
δ from some point on one of the other two sides. In [12, Proposition 7.4.7], the authors
show that an analogous version of Theorem 3.21 holds when S is a semigroup of isome-
tries of a Gromov hyperbolic metric space. The same paper also gives a generalisation of
Theorem 1.4 (see [12, Proposition 7.3.1]).
72 3. SEMIGROUPS THAT FIX THE UNIT CIRCLE
Let us now consider three special families of Mobius transformations in Aut(H2), namely
(i) z 7→ (az − b)/(bz + a), a2 + b2 = 1;
(ii) z 7→ λz, λ ≥ 1;
(iii) z 7→ z + µ, µ ≥ 0.
Each of these is a one-parameter semigroup. The first consists of all elliptic rotations
about i, the second consists of loxodromic transformations with attracting fixed point
∞ and repelling fixed point 0, and the third is a collection of parabolic transformations
that fix ∞ (and the identity is contained in each family too). We will prove that, up to
conjugacy, any closed semigroup that is not semidiscrete contains one of these families.
However, there is a caveat here: we must allow conjugacy not just by elements of Aut(H2),
but by Mobius transformations that transpose the upper and lower half-planes. Elements
of Aut(H2) are maps of the form z 7→ (az+b)/(cz+d), where a, b, c, d ∈ R and ad−bc > 0;
we also allow conjugation by maps of the same form but with negative determinant, that
is ad − bc < 0. We shall denote the collection of these maps by Aut(R), since each
such transformation fixes the extended real line. The reason we need to conjugate by
transformations in Aut(R), rather than just Aut(H2), is to simplify the treatment of case
(iii); after all, the two semigroups z 7→ z + µ | µ ≥ 0 and z 7→ z + µ | µ ≤ 0 are
conjugate by z 7→ −z, but they are not conjugate in Aut(H2). In the next lemma, and
indeed throughout this thesis, we write S to mean the closure of S in Aut(R3).
Lemma 3.22. Let S be a closed semigroup that is not semidiscrete. Then S is conjugate
by an element in Aut(R) to a semigroup that contains one of the families (i), (ii) or (iii).
Proof. As S is not semidiscrete, there is a sequence gn in S \ I that converges
uniformly to I, the identity. Suppose first that this sequence contains infinitely many
loxodromic elements. By passing to a subsequence, we can assume that every map gn is
loxodromic. Let αn and βn be the attracting and repelling fixed points of gn, respectively.
By passing to a further subsequence of gn, we can assume that the sequences αn and
βn both converge. Suppose for now that they converge to distinct values, which, after
conjugating S, we can assume are ∞ and 0, respectively. Let hn be any sequence of
Mobius transformations that satisfies hn(0) = αn, hn(∞) = βn and hn → I as n → ∞.
Define kn = h−1n gnhn; this map has repelling fixed point 0 and attracting fixed point ∞,
4. PROOF OF THEOREM 3.2 73
so kn(z) = λnz, where λn > 1. Furthermore, kn → I as n → ∞ (so λn → 1). Now select
any number λ > 1, and define f(z) = λz. By passing to yet another subsequence of gn
if necessary, we can assume that λn < λ for n = 1, 2, . . . . For each positive integer n,
the sequence λn, λ2n, λ
3n, . . . is strictly increasing with limit ∞. Define tn to be the unique
positive integer such that λ ∈ [λtnn , λtn+1n ). Let fn = ktnn . Notice that fn(1) = λtnn and
f(1) = λ. Also,
χ(fn(1), f(1)) ≤ χ(ktnn (1), ktn+1n (1)) ≤ σ(ktnn , k
tn+1n ) = σ(I, kn).
Since fn fixes 0 and ∞ for each n, we see that fn(1) → f(1), fn(0) → f(0) and
fn(∞) → f(∞) as n → ∞, so fn → f . Therefore, in this case, the family of type
(ii) is contained in our semigroup.
Near the start of the preceding argument we assumed that the sequences αn and βn
converged to distinct values. Let us resume the argument from that point, but this time
assume that αn and βn converge to the same value, which, after conjugating S if need
be, we can assume is ∞. Let hn be any sequence of Mobius transformations that satisfies
hn(∞) = βn and hn → I as n→∞. Define kn = h−1n gnhn; this map has attracting fixed
point∞. Let δn be the repelling fixed point of kn. Then δn →∞ as n→∞. By passing to
a subsequence of gn if need be, we can assume that the numbers δn all have the same sign,
which, after conjugating by z 7→ −z if need be, we can assume is negative. We can write
kn(z) = λn(z − δn) + δn, where λn > 1. As before, kn → I as n → ∞ (so λn → 1). Now
select any number µ > 0, and define f(z) = z+µ. By passing to yet another subsequence
of gn if necessary, we can assume that kn(0) = δn(1 − λn) < µ for n = 1, 2, . . . . For each
positive integer n, the sequence kn(0), k2n(0), k3n(0), . . . is strictly increasing with limit ∞.
Define tn to be the unique positive integer such that µ ∈ [ktnn (0), ktn+1n (0)). Let fn = ktnn .
Then
χ(fn(0), f(0)) ≤ χ(ktnn (0), ktn+1n (0)) ≤ σ(I, kn).
Hence fn(0)→ f(0); that is, δn(1−λtnn )→ µ. Since δn →∞, we see that λtnn → 1. Hence,
for any real number x,
fn(x) = λtnn x+ δn(1− λtnn )→ x+ µ as n→∞.
74 3. SEMIGROUPS THAT FIX THE UNIT CIRCLE
So fn → f . Therefore, in this case, the family of type (iii) is contained in our semigroup.
We began this proof by choosing a sequence gn in S \I such that gn → I, and supposing
that there were infinitely many loxodromic maps in this sequence. If there are infinitely
many parabolic maps in the sequence, then we can carry out an argument similar to those
we have given already to show that a conjugate of S contains the family of type (iii). The
remaining possibility is that almost all the maps gn are elliptic, in which case we may
as well assume that they are all elliptic. If any one of them is elliptic of infinite order,
then we obtain the family of type (i) (up to conjugacy) by considering the closure of the
semigroup generated by that map alone. If infinitely many of the maps gn share a fixed
point, then it is straightforward to see that S is conjugate to a semigroup that contains
the family of type (i). The only other possibility is that the maps gn have infinitely many
different fixed points. By passing to a subsequence of gn, we can assume that the fixed
points of gn are pairwise distinct. Let hn = gngn+1g−1n g−1n+1. Since the maps gn are of
finite order, hn is an element of S, and hn → I as n → ∞. Furthermore, one can easily
check that hn is loxodromic, so by the earlier arguments we see that the closure of our
semigroup contains one of the families of types (ii) or (iii).
For an elliptic map h ∈ Aut(H2) let fix(h) denote its unique fixed point in H2 and let
θ(h) ∈ [0, 2π) be its angle of rotation. Each of the families (i), (ii) and (iii) can by
parametrised by one real parameter. In the case of (i), we use the angle of rotation of
the elliptic map, while in (ii) and (iii) we use λ and µ respectively. In order to simplify
notation we let t denote the parameter in each family and let ft denote element in the
family, where it is understood that t ∈ R in family (i), t ≥ 1 in (ii) and t ≥ 0 in (iii).
Lemma 3.23. Suppose that g is a loxodromic Mobius transformation with 0 < αg < βg <
+∞. Then in each of the families (i), (ii) and (iii) there is an interval (a, b) ⊂ R such
that ftg is elliptic for all t ∈ (a, b). Moreover the functions t 7→ fix(ftg) and t 7→ θ(ftg)
are continuous on (a, b), and the former is injective.
Proof. There is an algebraic proof of this lemma, but the geometric proof we offer
is more illuminating. We consider cases (i), (ii) and (iii) in turn; first case (i). Let γ
denote the reflection in the hyperbolic line `γ that passes through i and is orthogonal to
4. PROOF OF THEOREM 3.2 75
the axis of g. Then g = γβ, where β is the reflection in another line `β that is parallel
to `γ . Let `t denote the line that passes through i such that ft = αtγ, where αt is the
reflection in `t. For some value of t, t = τ say (τ = π suffices), `τ intersects `β at a
single point in H2. Moreover, fτg = ατβ is elliptic, because `τ and `β intersect at a point
in H2. Since the map t 7→ ft is continuous, the Mobius group is a topological group,
and the set of elliptic transformations is an open subset of the Mobius group, it follows
that there is some open interval (a, b) ⊂ R containing τ such that ftg is elliptic for all
t ∈ (a, b). Moreover, since the functions h 7→ fix(h) and h 7→ θ(h) are continuous, it follows
that the maps t 7→ fix(ftg) and t 7→ θ(ftg) are also continuous. Indeed, the continuity
is geometrically clear since fix(ftg) is the point where `β and `t intersect and θ(ftg) is
twice the anticlockwise angle between `β and `t. It is also clear from the geometry that
both functions are injective, and we verify this for the function t 7→ fix(ftg). Suppose,
then, that a point ζ ∈ H2 is the common fixed point of ftg and fsg, for some t, s ∈ (a, b).
This means that ft−s fixes g(ζ). If t 6= s then we must have g(ζ) = i, which is not pos-
sible since ζ ∈ `β, i ∈ `γ and g(`β) does not meet `γ . Hence g(ζ) 6= i, and so t = s after all.
Now for case (ii). This time let `γ be the line that is orthogonal to the axis of g and to the
vertical hyperbolic line L between 0 and ∞ (the positive imaginary axis). In Euclidean
terms, `γ is a Euclidean semicircle centred on the origin that is symmetric about L. Out-
side this semicircle is contained the hyperbolic line `β that is orthogonal to the axis of g
and satisfies g = γβ (where β and γ are the reflections in `β and `γ respectively). Let αt
denote the reflection in the line `t, chosen such that `t is orthogonal to L and such that
ft = αtγ. Note that ft, which belongs to family (ii), has attracting fixed point ∞ and
repelling fixed point 0, and so for t > 0 the line `t is outside the semicircle bounded by
the real axis and `γ . Note that `β is not orthogonal to L, since `γ is the unique geodesic
that is orthogonal to both L and the axis of g. It follows that we can choose a value
of t, t = τ say, such that `t meets `β at exactly one point in H2. Hence at t = τ , the
transformation ftg = αtβ is elliptic and fixes the point in the hyperbolic plane where `t
cuts `β. As in the case of family (i), we can choose an open interval (a, b) ⊂ R+ containing
τ such that ftg is an elliptic transformation for all t ∈ (a, b). Again, since the functions
76 3. SEMIGROUPS THAT FIX THE UNIT CIRCLE
h 7→ fix(h) and h 7→ θ(h) are continuous, the functions t 7→ fix(ftg) and t 7→ θ(ftg) are
also continuous on (a, b). The former function is injective, for if a < s ≤ t < b and
fsg and ftg both fix the same point ζ ∈ H2, then f−1s ft = ft−s fixes g(ζ). If t 6= s then
ft−s is loxodromic and so cannot fix any point in H2, hence we must have t = s, as required.
Case (iii) is similar to case (ii), and we omit the argument.
We now set about proving Theorem 3.2. We will need the following result of Barany,
Beardon and Carne [2, Theorem 3], mentioned in the introduction.
Theorem 3.24. Suppose that f and g are noncommuting elements of Aut(D), and one of
them is an elliptic element of infinite order. Then 〈f, g〉 is dense in Aut(D).
We also need the following lemma, which states two separate results. The first follows
from the characterisation of elementary groups given on page 84 of [3], while the second
follows from [3, Theorem 8.4.1].
Lemma 3.25.
(i) If an elementary group contained in Aut(D) contains elliptic transformations, then
they all fix a common point in D.
(ii) If a nonelementary subgroup of Aut(D) contains elliptic transformations whose fixed
points accumulate in D, then the group contains an elliptic transformation of infinite
order.
The next theorem is a stronger version of Theorem 3.2 (we will need this stronger statement
later). Recall from the introduction that Mob(J) denotes the semigroup of those Mobius
transformations that map a closed interval J within itself.
Theorem 3.26. Suppose that S is a semigroup that is not elementary, semidiscrete, or
contained in Mob(J), for any nontrivial closed interval J ⊂ S1. Then there is a two-
generator semigroup within S that is dense in Aut(D).
Proof. We frame the proof using the upper half-plane model. By Lemma 3.22, we
can assume, after conjugating S by a Mobius transformation in Aut(R), that its closure
4. PROOF OF THEOREM 3.2 77
S contains one of the families of maps (i), (ii) or (iii). In order to apply Lemma 3.23, we
will prove that there is a loxodromic map g in S with 0 < αg < βg < +∞ (possibly after
conjugating S again). To this end, suppose first that S contains the family of type (i).
The sets Λ−(S) and Λ+(S) are perfect, and, since S is not elementary, by Theorem 3.21
we can choose a loxodromic map g in S such that χ(αg, βg) is less than χ(0,∞) = 2, the
maximum value of χ. After conjugating S by elliptic rotations about i, and possibly the
map z 7→ −z (which fixes the family of type (i)), we can assume that 0 < αg < βg < +∞.
Suppose now that S contains a family of type (ii); then 0 ∈ Λ−(S) and ∞ ∈ Λ+(S).
One possibility might be that Λ−(S) is contained in a closed interval J , and Λ+(S) is
contained in the closed interval R \ J . In fact this cannot be: for then there exists a
unique complementary connected component of Λ− whose interior meets Λ+. Let us
denote this component, regarded as a subset of R, by U . Since S is not elementary, U has
a non-empty interior and U 6= R. Since elements of S map complementary components of
Λ− into complementary components of Λ−, we see that U is forward invariant under S,
and so S ⊂ Mob(U) contrary to our assumption. It follows, then, that there exist points
p in Λ−(S) and q in Λ+(S) with either −∞ < p < q < 0 or 0 < q < p < +∞. After
conjugating S by the map z 7→ −z (which fixes the family of type (ii)), we can assume
that the points p and q satisfy 0 < q < p < +∞. We can now apply Theorem 3.21 to
deduce the existence of a loxodromic map g in S with 0 < αg < βg < +∞.
The remaining case is that S contains a family of type (iii). This can be dealt with in
a similar way to the previous case, only this time if we have −∞ < p < q < 0 and not
0 < q < p < +∞, we must conjugate by a parabolic map of the form z 7→ z + t (for some
t ∈ R) that fixes the family (iii). Once again we deduce the existence of a loxodromic map
g with 0 < αg < βg < +∞.
We are now in a position to apply Lemma 3.23, which asserts the existence of an interval
(a, b) ⊂ R such that ftg is elliptic for all t ∈ (a, b). Moreover there is a dense subset
A ⊂ (a, b) such that ft ∈ S for all t ∈ A. One possibility is there exists t ∈ A such that
ftg is an elliptic map of infinite order. In this case Theorem 3.24 tells us that 〈ftg, g〉, and
hence S, is dense in Aut(H2). Otherwise, the collection of functions F = ftg | t ∈ A are
elliptic elements of finite order within S. By Lemma 3.23 the fixed points of F accumulate
78 3. SEMIGROUPS THAT FIX THE UNIT CIRCLE
in H2. Hence the group generated by F is contained in S, and by Lemma 3.25 (i) this
group is not elementary. Hence by Lemma 3.25 (ii) the group generated by F contains
an elliptic element of infinite order, h say. Now Theorem 3.24 that tells us that 〈h, g〉 is
dense in Aut(H2), and so S is dense in Aut(H2), as required.
Our next objective is to obtain a version of Theorem 3.2 for finitely-generated semigroups,
as promised at the beginning of this chapter. However, before we do that, we must take
a detour to study some anomalous semigroups that are exceptional cases in the version of
Theorem 3.2 for finitely-generated semigroups and in Theorem 3.3.
5. Classification of finitely-generated semigroups and proof of Theorem 3.3
We now take a closer look at exceptional semigroups, given on page 10 in the introduction.
Recall that a semigroup S of Mobius transformations is exceptional if it lies in Mob(J) for
some nontrival closed interval J , and the collection of elements of S that fixes J as a set
forms a nontrivial discrete group, and there is another element of S outside this discrete
group that fixes one of the end points of J . One of the reasons that exceptional semigroups
are special is that, although finitely-generated semigroups contained in Mob(J) are usually
semidiscrete, exceptional semigroups are never semidiscrete.
Lemma 3.27. Exceptional semigroups are not semidiscrete.
Proof. By conjugating, we may assume that J = [0,+∞] and the group part of S
comprises maps of the form gn(z) = λnz, n ∈ Z, where λ < 1. Furthermore, we may
assume that there is a map f(z) = az + b in the inverse free part of S, where a > 0 and
b > 0. Observe that, since f ∈ S \ S−1, the maps
gnfg−1n (z) = az + λnb
lie in S \ S−1 for each n ∈ N. Therefore the map h(z) = az lies in the closure S \ S−1 of
S \ S−1 in Aut(D). By composing h with the maps gn we see that the identity belongs to
S \ S−1, and so S is not semidiscrete.
5. CLASSIFICATION OF FINITELY-GENERATED SEMIGROUPS AND PROOF OF THEOREM 3.3 79
In the remainder of this section we explain how it is straightforward to identity a finitely-
generated exceptional semigroup by examining the generating set alone. We need the
following lemma.
Lemma 3.28. Suppose that S is a semigroup contained in Mob(J), for some nontrivial
closed interval J , that is generated by a set F . Then S has an element that fixes exactly
one of the end points of J if and only if F contains such an element.
Proof. If F has an element that fixes exactly one of the end points of J , then certainly
S does too, as F ⊆ S. Conversely, suppose that no element of F fixes exactly one of the
end points of J . Suppose that an element f of S fixes one of the end points a of J . We
can choose maps fi in F such that f = f1 · · · fn. Observe that f(J) ⊆ f1(J), so a ∈ f1(J),
which implies that f1(a) = a. Therefore the element f2 · · · fn of S fixes a, and, by our
assumption, f1(J) = J . Repeating this argument we see that fj(J) = J for j = 1, . . . , n,
so f(J) = J . We have now proved that no element of S fixes exactly one end point of J ,
as required.
Suppose now that we have a semigroup S generated by a finite set F , and we wish to
know whether S is exceptional. First look for a collection of two or more loxodromic
transformations in F with the same axis. Lemma 3.6 tells us precisely when this collection
of loxodromic maps generates a discrete group. If there is no such collection, or more than
one such collection, then S is not exceptional. Let us suppose there is indeed one such
collection, with axis γ, and this collection does generate a discrete group. In order for
S to be exceptional, every element of F must lie in Mob(J), where J is one of the two
intervals in R with the same end points as γ. If this is the case, then Lemma 3.28 tells
us that S is exceptional if and only if F contains an element that fixes exactly one of the
end points of J .
5.1. A classification of finitely-generated semigroups. Here we prove Corol-
lary 3.30, which is a version of Theorem 3.2 for finitely-generated semigroups. Given a
closed interval J , let us denote by Mob0(J) the group part of Mob(J), which comprises
those elements in Mob(J) that fix J as a set. It is the one-parameter family of loxodromic
80 3. SEMIGROUPS THAT FIX THE UNIT CIRCLE
Mobius transformations whose fixed points are the end points of J . If J = [0,+∞], then
Mob0(J) consists of all maps of the form z 7→ λz, where λ > 0.
Theorem 3.29. Let S be a finitely-generated semigroup that is contained in Mob(J) for
some nontrivial closed interval J . Then S is semidiscrete if and only if it is nonexceptional
and not dense in Mob0(J).
Proof. By Lemma 3.27, exceptional semigroups are not semidiscrete, and, obviously,
nor are semigroups that are dense in Mob0(J). Suppose now that S is neither exceptional
nor dense in Mob0(J). By conjugation, we may assume that J = [0,+∞]. Let us denote
by T the collection of maps in S that fix J as a set. Each element of T has the form
z 7→ λz, where λ > 0. We split the remaining argument into two cases depending on
whether (i) T is a nontrivial discrete group, or (ii) all elements z 7→ λz of T satisfy λ ≥ 1,
or else they all satisfy λ ≤ 1. We also include in (ii) the possibility that T = ∅. Let F be
a finite generating set for S.
In case (i), as S is not an exceptional semigroup, there is an interval K = [a, b], where
0 < a < b < +∞, such that if f ∈ F and f does not fix J as a set, then f(J) ⊆ K.
Suppose now that Fn is a sequence of distinct elements of S. We wish to show that Fn
cannot converge to the identity, because doing so will demonstrate that S is semidiscrete.
If Fn ∈ T for infinitely many n ∈ N, then certainly Fn cannot converge to the identity
because T is discrete. Otherwise, when n is sufficiently large, we can write Fn = GnfnHn,
where Gn ∈ T (possibly Gn = I), fn ∈ F \ T and Hn ∈ S. Notice that
fnHn(J) ⊆ fn(J) ⊆ K.
Let Gn(z) = λnz, where λn > 0. Either λn ≥ 1, in which case Fn(0) ≥ fnHn(0) ≥ a, or
λn ≤ 1, in which case Fn(∞) ≤ fnHn(∞) ≤ b. Therefore Fn does not converge to the
identity.
Let us move on to case (ii), and let us assume that every element of T has the form
z 7→ λz, where λ ≤ 1 (the other case is similar). We include the possibility that T = ∅
in our analysis. Let D denote the open top-right quadrant of the complex plane. This
is the hyperbolic half-plane in H2 with boundary J . There exists a Euclidean line ` that
5. CLASSIFICATION OF FINITELY-GENERATED SEMIGROUPS AND PROOF OF THEOREM 3.3 81
intersects R at some point x < 0 and has a positive slope chosen such that it intersects D
with the property that for any loxodromic element of F that has αf = ∞ and f(0) > 0,
we have βf < 0, and for any element f ∈ F \T that does not fix ∞, the domain f(D) lies
entirely below `. This is possible since F is finite. Let U denote the subset of D consisting
of points below the line `.
It is easy to check that each element in F \ I maps U strictly inside itself. Since each
f ∈ F ∩ T maps ` to a parallel Euclidean line strictly below `, we see each such f maps
U strictly inside itself. If f ∈ F \ T and f does not fix ∞, then f maps D to a Euclidean
semicircle bounded by a hyperbolic geodesic, both of whose landing points lie in [0,+∞).
We choose the slope of ` so that each such semicircle is contained in U . If f ∈ F \ T and
f(∞) =∞, then possibly f is parabolic, in which case f(z) = z+ c for some c ∈ (0,+∞),
and so f maps U strictly inside itself. Otherwise f is loxodromic, and either it has at-
tracting fixed point ∞ and repelling fixed point in (−∞, 0), in which case by choice of x
it can be seen that f maps U strictly inside itself; or it has repelling fixed point ∞ and
attracting fixed point in (0,+∞), and the same conclusion can be drawn in this case too.
Now for each f ∈ F \ I we can choose a point zf ∈ H2 \ U such that f(zf ) ∈ U . Any
element of S \ I maps at least one point of the finite set zf | f ∈ F \ I into U . To
see this, express the given element as a word in the generating set F \ I and consider
the action of the element on zf , where f is the rightmost term in this word. It follows
that S is semidiscrete.
We can now state a version of Theorem 3.2 for finitely-generated semigroups, which follows
immediately from Theorems 3.2 and 3.29.
Corollary 3.30. Let S be a finitely-generated semigroup. Then S is either
(i) elementary; or
(ii) semidiscrete; or
(iii) contained in Mob(J), for some nontrivial closed interval J , and is either exceptional
or dense in Mob0(J); or
(iv) dense in Aut(D).
82 3. SEMIGROUPS THAT FIX THE UNIT CIRCLE
5.2. Proof of Theorem 3.3. Our next task is to prove Theorem 3.3. This theorem
is the semigroups counterpart of a well-known result (see for example [3, Theorem 5.4.2])
in the theory of Fuchsian groups, which says that a nonelementary group of Mobius trans-
formations is discrete if and only if every two-generator subgroup is discrete. Here is a
version of that result for semigroups, which, unlike Theorem 3.3, does not assume that
the semigroup is finitely-generated.
Theorem 3.31. Let S be a nonelementary semigroup that is not contained in Mob(J), for
any nontrival closed interval J . Then S is semidiscrete if and only if every two-generator
semigroup contained in S is semidiscrete.
This theorem is an immediate corollary of Theorem 3.26. The assumption that S is not
contained in Mob(J) cannot be removed. To see why this is so, consider, for example,
the semigroup S generated by the maps (1 − 1/n)z for n = 2, 3, . . . . Clearly S is not
semidiscrete, but every two-generator semigroup within S is semidiscrete. This particular
semigroup is elementary, but it is easy to adjust it to give a nonelementary example:
simply append to S any element of Mob([0,+∞]) that fixes neither 0 nor ∞.
Let us now turn to Theorem 3.3, which states that any finitely-generated nonexceptional
semigroup S contained in Aut(D) is semidiscrete if and only if every two-generator semi-
group contained in S is semidiscrete.
Proof of Theorem 3.3. If S is semidiscrete, then certainly any two-generator semi-
group in S is semidiscrete. Conversely, suppose that every two-generator semigroup con-
tained in S is semidiscrete. By Theorem 3.26, S is either elementary, semidiscrete or
contained in Mob(J), for some closed interval J . If S elementary and is not contained in
the stabiliser group of z, for some z ∈ C, then one can check each case in Theorem 3.5 and
verify that S semidiscrete. If S is contained in Mob(J) and is not elementary or semidis-
crete, then Corollary 3.30 tells us that S is dense in Mob0(J). However, Lemma 3.6(iii)
shows that this cannot be so; therefore S cannot be contained in Mob(J) but nonelemen-
tary or semidiscrete after all. We conclude that, in each case, S is semidiscrete.
6. INTERSECTING LIMIT SETS 83
6. Intersecting limit sets
In this section we prove Theorem 3.4, which says that if S is a nonelementary, finitely-
generated and semidiscrete semigroup contained in Aut(D), then Λ+(S) = Λ−(S) if and
only if S is a group. The next lemma is an important step in establishing this result.
Lemma 3.32. Let S be a nonelementary semigroup contained in Aut(B3) that satisfies
Λ−(S) ⊆ Λ+(S). Let f ∈ S, p ∈ Λ−(S), and let U be an open neighbourhood of p. Then
there exists an element g of S such that fg is loxodromic with attracting fixed point in U .
Proof. By Theorem 3.21, we can choose a loxodromic element h1 of S such that
βh1 ∈ U . Now observe that for any point ζ in B3, the sequence (fhn1 )−1(ζ) converges
ideally to p. By replacing h1 with a power of h1 if necessary, we can assume that fh1 is
loxodromic, and using the observation that we just made we can assume that βfh1 ∈ U .
Let V be an open neighbourhood of βfh1 such that fh1(V ) ⊆ U . Appealing to Theo-
rem 3.21 again, we can choose a loxodromic element h2 of S such that αh2 ∈ V . By
replacing h2 by a power of h2 if necessary we can ensure that, first, h2(U) ⊆ V and,
second, fh1h2 is loxodromic. Since fh1h2(U) ⊆ fh1(V ) ⊆ U we see that αfh1h2 ∈ U . The
lemma now follows on choosing g = h1h2.
The main theorem of this section follows.
Theorem 3.33. Let S be a finitely-generated, nonelementary semidiscrete subsemigroup
of Aut(D) that satisfies Λ−(S) ⊆ Λ+(S). Then S is a group.
Proof. Let g ∈ S. Choose two distinct points u and v in Λ−(S). By Lemma 3.32 it
is possible to construct a composition sequence Fn = f1 · · · fn generated by S that accu-
mulates at both u and v. To see this, observe that given Fn = f1 · · · fn such that Fn(0) is
within ε (with respect to the chordal metric) of u, we can, by Lemma 3.32, choose h ∈ S
such that αFnh is within ε/2 of v. Then for all large enough k ∈ N, the point (Fnh)k(0)
is within ε of v; hence we can choose fn+1 = h(Fnh)k−1 such that Fn+1(0) is within ε
of v. Furthermore, we can easily fashion this composition sequence such that fi = g for
infinitely many positive integers i. Notice that Fn is discrete, because S is semidiscrete,
84 3. SEMIGROUPS THAT FIX THE UNIT CIRCLE
by Theorem 2.8.
If Fn is an escaping sequence, then S is a group, by Theorem 3.9. If the sequence Fn
is not an escaping sequence then because it is discrete, it has a constant subsequence
Fn1 , Fn2 , . . . . Choose m > n1 such that fm = g, and choose k such that nk < m ≤ nk+1.
Then
fnk+1fnk+2 · · · fnk+1= F−1nk
Fnk+1= I.
Hence g−1 ∈ S. As we chose g arbitrarily from S, we conclude that S must be a group.
Theorem 3.4 given at the start of this chapter is an immediate consequence of this the-
orem. By referring to Section 1 of this chapter we see that the only finitely-generated
semidiscrete semigroups that do not contain loxodromic elements are either finite groups
of elliptic maps with a common fixed point, or semidiscrete semigroups of parabolic maps
with a common fixed point. All these semigroups satisfy Λ− ⊆ Λ+, but some of the latter
type are not groups, such as 〈z 7→ z + 1〉.
Theorem 3.33 only applies to subsemigroups of Aut(D); however, provided that we addi-
tionally suppose the forward limit set of S is not connected, its proof can be adapted for
subsemigroups of Aut(B3). First we give a preparatory lemma.
Lemma 3.34. Suppose F is a bounded collection of elements in Aut(B3). If Fn is an
escaping composition sequence generated by F , then Λ+(Fn) is connected.
Proof. Suppose Fn = f1 · · · fn where fn belongs to F for each n ∈ N. Choose any
ε > 0. Since F is bounded in Aut(B3), for any ζ ∈ B3 there is some positive real number M
such that ρ(ζ, f(ζ)) ≤ M for each f ∈ F . Hence ρ(Fn(ζ), Fn+1(ζ)) = ρ(ζ, fn+1(ζ)) ≤ M
for all n. Since Fn(ζ) accumulates only on the ideal boundary and ρ(Fn(ζ), Fn+1(ζ)) ≤M ,
we have χ(Fn(ζ), Fn+1(ζ)) < ε for all large enough n. Now suppose towards contradiction
that Λ+(Fn) is not connected. Then we can choose two open subsets of S2, U and V , both
of which meet Λ+(Fn), that are a positive distance apart, and are such that Λ+(Fn) ⊆
U∪V . It follows that for all large enough n we have χ(Fn(ζ), U∪V ) < ε. Hence can choose
6. INTERSECTING LIMIT SETS 85
positive integers p and q, where p < q, and such that both χ(Fp(ζ), U) and χ(Fq(ζ), V )
are less than ε, and that satisfy χ(Fn(ζ), Fn+1(ζ)) < ε and χ(Fn(ζ), U ∪ V ) < ε for each
n = p, p + 1, . . . , q − 1. By choosing ε small enough, for example ε < 14χ(U, V ), this
contradicts the fact that χ(Fn(ζ), U ∪ V ) < ε for each n = p, p+ 1, . . . , q − 1.
Theorem 3.35. Let S be a nonelementary, finitely-generated semidiscrete subsemigroup
of Aut(B3) that satisfies Λ−(S) ⊆ Λ+(S). If Λ+(S) is not connected then S is a group.
Proof. Suppose towards contradiction that Λ− is contained in one connected com-
ponent of Λ+, which we shall denote by Λ+0 . Let U ⊆ S2 be any open set that meets Λ+.
By Theorem 1.8(i) applied to S−1 we can choose a loxodromic transformation g ∈ S with
attracting fixed point in U . Since Λ− is contained in Λ+0 , the repelling fixed point of g,
namely βg, must lie in Λ+0 . Since each element of S maps any component of Λ+ within
another component, it follows that g must map Λ+0 within itself. Since S is nonelementary,
Λ+ is a perfect set, so Λ+0 contains points other than βg. Hence for all large enough n,
the set gn(Λ+0 ) meets U . Since U was arbitrary, Λ+
0 must be the only component of Λ+,
contradicting our assumption that Λ+ is not connected. It follows that we can assume Λ−
is not contained within a single connected component of Λ+.
Now just as in the proof of Theorem 3.33, for any pair of distinct points u and v in Λ−,
which we take to lie in different components of Λ+, we can find a composition sequence
Fn = f1 · · · fn that accumulates at both points. (Strictly speaking, Theorem 3.33 assumes
S ⊆ Aut(D), but just the same argument works here.) Moreover, for any g ∈ S we can
choose Fn such that fi = g for infinitely many i ∈ N. If Fn is an escaping sequence then
by Lemma 3.34 Λ+(Fn) is connected, and so Λ+ must have a connected component that
contains u and v, which contradicts the fact that u and v lie in different components of Λ+.
Hence Fn cannot be an escaping sequence, and because Fn is discrete it has a constant
subsequence Fn1 , Fn2 , . . . . Choose m > n1 such that fm = g, and choose k such that
nk < m ≤ nk+1. Then
fnk+1fnk+2 · · · fnk+1= F−1nk
Fnk+1= I.
Hence g−1 ∈ S. As we chose g arbitrarily from S, we conclude that S must be a group.
86 3. SEMIGROUPS THAT FIX THE UNIT CIRCLE
We suspect the assumption that Λ+(S) is not a connected set in Theorem 3.35 can be
replaced with the assumption that Λ+(S) 6= C. Whether or not this can be done is an
open problem under investigation. Nevertheless, it can be shown that if there exists a
nonelementary finitely-generated semidiscrete semigroup S whose forward and backward
limit sets are equal to the connected set Λ 6= C, then S is contained in a Kleinian group
with limit set Λ.
CHAPTER 4
Limit sets
1. Introduction
In this chapter we study the limit sets of a semigroup, their internal structure, and what
limit sets can tell us about the semigroup that generates them. We have already utilised
analogues of the conical and horospherical limit sets from the theory of Kleinian groups
in Chapter 3. In Section 2 we introduce further noteworthy subsets of the limit sets and
study their properties. In Section 3 we study semigroups whose limit sets are disjoint,
and expand on recent work of Fried, Marotta and Stankewitz [15]. Finally, we consider
how the limit sets of a semigroup behave under perturbation of its generators. Most of
the results in this chapter relate to semigroups in any number of dimensions, but are not
as strong as those obtained in Chapter 3 on semigroups acting on two dimensions.
We next give two lemmas from [15], which for completeness we record here.
Lemma 4.1. (Topological transitivity)
Let S be a semigroup and suppose U ⊆ S2 is any open set that meets Λ−. Then S(U)
is either the whole ideal boundary, or the whole ideal boundary missing one point. In
the latter case, the missing point is fixed by each element in S, and in particular S is
elementary.
Proof. First note that S(U) is forward invariant with respect to S. By Theorem 1.8
(ii) we see that S is not a normal family on U . Hence by Montel’s theorem, S(U) misses
at most two points on C. First suppose towards contradiction that S(U) misses exactly
two points, say w and z. Then since S(U) is forward invariant, its complement, w, z,
is backward invariant. This means that w, z is a finite orbit of S, and in particular, S
is elementary. By consulting Proposition 1.7 and its proceeding discussion, we see that
S belongs either to case (c) or to case (f) given in the Proposition. By checking these
87
88 4. LIMIT SETS
possibilities it can be seen that in fact S(U) can miss at most one point, contrary to our
assumption. Hence S(U) misses at most one point in S2, z say. Then it follows that S
fixes z, as the complement of the forward invariant set S(U).
Lemma 4.2. Suppose S is a semigroup and z is a point in Λ− that is not fixed by every
element in S. Then Λ− = S−1(z). Similarly if z ∈ Λ+ is not fixed by every element in S,
then Λ+ = S(z).
Proof. Pick any ζ ∈ Λ−, and let U be any open neighbourhood of ζ. Then by
Theorem 4.1 we have z ∈ g(U) for some g ∈ S, that is g−1(z) ∈ U . Since U was arbitrary,
ζ is an accumulation point of S−1(z), and so Λ− ⊆ S−1(z). Conversely, since z ∈ Λ− and
Λ− is backward invariant, we see that S−1(z) ⊆ Λ−. Taking closures we obtain the reverse
inclusion. Applying the same argument to S−1 gives the second statement.
As we are studying limit sets it is convenient to introduce notation to represent their
complements. Accordingly, for a semigroup S we define Ω+(S) = S2 \Λ+(S) and Ω−(S) =
S2 \ Λ−(S). As usual, if there is no danger of ambiguity we suppress mention of S.
2. On the structure of the limit sets
2.1. The sets D+ and D−. Here we introduce two important subsets of the limit
sets of a semigroup. For any semigroup S we define the sets
D−(S) =z ∈ Λ−(S) | Λ−(S) ⊆ S(z)
and
D+(S) =z ∈ Λ+(S) | Λ+(S) ⊆ S−1(z)
.
As usual we shall simply write D+ or D− if the semigroup is clear from the context. We
first note that D+ is a forward invariant set and D− is backward invariant. When S is a
nonelementary group, the sets D+ or D− are both equal to Λ+ = Λ−, and so any utility
of these sets can be realised only when S is not a group – there is no useful analogue of
D+ or D− in the theory of Kleinian groups.
By Lemma 4.2 for any z ∈ Λ+, the closure of the orbit of S(z) is equal to Λ+. If moreover
z lies in D+, then the closure of the orbit of S−1(z) covers Λ−. Recall that if f is a rational
2. ON THE STRUCTURE OF THE LIMIT SETS 89
function acting on C then the Julia set J(f) is defined as the set of non-normality of the
family fn | n ∈ N. The following result (sometimes called topological transitivity) is
well known: for any rational function f the set of points z ∈ J(f) for which the orbit of z
is dense in J(f), is itself dense in J(f). The following theorem is motivated by the proof
of that result, as given in [32, Corollary 4.16].
Theorem 4.3. Suppose S is a semigroup such that no point in Λ− is fixed by every element
in S. Then D− is a dense Gδ subset of Λ− with respect to the subspace topology on Λ−.
Proof. Working in the chordal metric, let Uj,k | k = 1, . . . , n(j) be an open cover
of Λ−, where Uj,k has chordal diameter less than 1/j and Uj,k meets Λ− for each j ∈ N
and each k = 1, . . . , n(j). Let Vj,k = S−1(Uj,k) for each j ∈ N and k = 1, . . . , n(j). Since
Uj,k meets Λ− and (by Lemma 4.2) Λ− = S−1(z) for all z ∈ Λ−, we can pick z ∈ Uj,k ∩Λ−
so that as
S−1(z) ⊆ S−1(Uj,k)
it follows that
S−1(z) ∩ Λ− ⊆ S−1(Uj,k) ∩ Λ−,
and since S−1(z) is a dense subset of Λ− we have
Λ− ⊆ S−1(Uj,k) ∩ Λ−.
Hence Vj,k ∩ Λ− is dense in Λ− for each j ∈ N and k = 1, . . . , n(j). As a complete metric
space, Λ− is a Baire space, that is, any countable intersection of open dense subsets of Λ−
is dense in Λ−. It follows that
D =⋂j∈N
k≤n(j)
(Vj,k ∩ Λ−) = Λ− ∩⋂j∈N
k≤n(j)
Vj,k
is dense in Λ−. Now if z ∈ D then for all j ∈ N and k ∈ 1, . . . , n(j) there is some g ∈ S
such that g(z) ∈ Uj,k. Hence S(z) has points arbitrarily close to any point in Λ−, and
so S(z) covers Λ−. This proves that D is a dense Gδ set with respect to the subspace
topology on Λ−, and that D ⊆ D−.
90 4. LIMIT SETS
By unpacking the definition, we verify that D− ⊆ D: For suppose z ∈ D−; since each
open set Uj,k meets Λ− and S(z) is dense in Λ−, S(z) meets each Uj,k. In other words
there exists some g ∈ S such that z ∈ g−1(Uj,k), so that z ∈ S−1(Uj,k). Since this holds
for each Uj,k and z lies Λ−, it follows that z lies in D− by definition.
Corollary 4.4. The set D− meets Λ+ if and only if Λ− ⊆ Λ+.
Proof. Clearly if Λ− ⊆ Λ+ then D− meets Λ+. Conversely suppose z lies in D−∩Λ+;
then we have
Λ− ⊆ S(z) = Λ+,
as required.
Recall that a subset of a topological space is meagre if it is the countable union of nowhere
dense sets. A residual set is the complement of a meagre set. Meagre sets can be regarded
as topologically small sets – the family of meagre sets form a σ-ideal, that is the family
contains all subsets and all countable unions of its elements. In a complete metric space
such as Λ−, a set is residual if and only if it contains a dense Gδ set. This means we can
think of D− as a topologically large set, at least with respect to the subspace topology
on Λ−. It follows from Theorem 4.3 and Corollary 4.4 that unless Λ− is contained in Λ+,
then Λ+ ∩ Λ− is a meagre subset of Λ−. Similarly, unless Λ+ is contained in Λ−, then
Λ− ∩ Λ+ is a meagre subset of Λ+. Recall that in dimension two, by Theorem 3.33, if
S is finitely-generated, nonelementary, semidiscrete and either limit set is contained in
the other, then S is a group. Hence we can restate Theorem 3.33 as follows: a finitely-
generated, nonelementary and semidiscrete subsemigroup of Aut(D) is a group if and only
if D+ meets D−, in which case D+ = D−.
Figure 4.1 shows the limit sets of the semigroup S generated by the maps f(z) =a
1 + z,
g(z) =a− 1 + 2ia1/2
1 + zand h(z) =
−0.25
1 + z, where a = −0.1 + 0.7i. These maps are chosen
such that f , gh and hg are parabolic elements. The forward limit set is shown in red,
while the backward limit set is shown in blue. The intersection Λ+(S) ∩ Λ−(S) is the set
of fixed points of f , gh and hg, which is certainly a meagre subset of both Λ+(S) and
Λ−(S).
2. ON THE STRUCTURE OF THE LIMIT SETS 91
Figure 4.1. Limit sets of a semigroup whose intersection is finite
The next theorem shows that if Λ− * Λ+, then D− ⊆ Λ−c . We know that if Λ− * Λ+
then D− does not meet Λ+; however there are many such semigroups for which Λ−c meets
Λ+. For example, take a Schottky subgroup G of Aut(D) and choose two disjoint closed
discs I− and I+ that are orthogonal to the unit disc and that do not meet the Dirichlet
region of G centred at 0. Choose an element f in Aut(D) that maps the interior of the
complement of I+ onto I−. The semigroup S generated by G ∪ f is semidiscrete, and
has group part G strictly contained in S. As Schottky groups are convex-cocompact, that
is the (hyperbolic) convex hull of the limit set intersected with any Dirichlet region is a
bounded set with respect to the hyperbolic metric, the conical limit set of G is the limit
set of G. Hence Λ−c (S) meets Λ+(S) on at least the limit set of G, however Λ−(S) * Λ+(S).
Theorem 4.5. Suppose S is a semigroup such that Λ− * Λ+; then D− ⊆ Λ−c .
92 4. LIMIT SETS
Proof. It follows from Corollary 4.4 that D− ∩ Λ+ = ∅. Now choose z ∈ D−, so by
definition of D− there exists a sequence gn ∈ S such that gn(z) converges to z. It follows
that z ∈ Λ−c (gn) for otherwise gn(z) must accumulate exactly where gn(ζ) accumulates for
any point ζ in hyperbolic space. This set of accumulation points is contained in Λ+, and
so cannot include z as D− is disjoint from Λ+. Hence z is a backwards conical limit point
of the sequence gn, and so lies in the backward conical limit set of S.
It follows from the theorem above that provided Λ− * Λ+, then Λ− \ Λ−c is a meagre
subset of Λ−. Notice that this holds whether or not S is discrete or semidiscrete, and
echoes the situation for a geometrically finite discrete group G, for which Λ(G) \Λc(G) is
a countable set.
The semigroup of Theorem 3.11 is an example of a semigroup for which D+ is not equal
to Λ+c , as we now explain. Consider (see Figure 4.2 below) the semigroup S = 〈g, h〉 where
g, h are loxodromic maps such that αg = βh but βg and αh are distinct points.
g h
αg = βh
αhβg
Figure 4.2. The axes of g and h
Then the closed interval contained in the unit circle that is separated from the axis of h
by the axis of g is equal to the backward conical limit set. Similarly the closed interval
contained in the unit circle that is separated from the axis of g by the axis of h is equal
to the forward conical limit set. Hence the point αg = βh lies in both conical limit sets,
and Λ+ is the proper closed interval [βh, αh]. As S fixes αg = βh, this point cannot lie in
either D+ or D−, because the S−1 orbit of αg = βh cannot accumulate everywhere in Λ+.
2. ON THE STRUCTURE OF THE LIMIT SETS 93
Similarly αh /∈ D+ because αh ∈ Ω+ and Ω+ is a backward invariant set, so that S−1(αh)
cannot accumulate in the interior of Λ+.
Before introducing the next interesting subset of Λ+, we give an explicit description of
D+(S) when S is the Cantor semigroup, that is S = 〈f0, f2〉 where fi(z) = 13(z + i) for
i = 0, 2. We shall shortly (in subsection 2.3) show that D+(S) is the set of points in Λ+(S)
whose base-3 expansion contains every finite string in 0, 2. Since the limit sets of S are
disjoint the forward conical limit is the full limit set, and so D+(S) is strictly contained
in Λ+c (S) in this case.
2.2. The point transitive limit set and an inclusion theorem. The forward
point transitive limit set of S, which we shall denote by Λ+p (S), is the set of points z on
the ideal boundary with the following property. For any ζ in hyperbolic space and half-
line γ landing at z, there exists an escaping sequence gn ∈ S such that ρ(gn(ζ), γ) → 0
as n → ∞. It is easy to see that Λ+p (S) is forward invariant, and so for nonelementary
semigroups Λ+p (S) is either empty or dense in Λ+. The backward point transitive limit set,
denoted by Λ−p (S), is defined as Λ+p (S−1). We remark that Λ+
p , and indeed Λ+c , are Gδσ
sets, as can be seen from the set-theoretic descriptions of Λ+p and Λ+
c given, for example,
in Nicholls [34, Theorem 2.4.3]. The point transitive limit sets of Kleinian groups are
utilised in number theory, see for example Lehner [25, Chapter 10]. Here we make little
use of them; however they have a pleasing geometric definition and we can relate them to
the other sets introduced thus far.
Theorem 4.6. Suppose S is a semigroup such that Ω− is not empty; then Λ+p ⊆ D+.
Proof. Suppose z ∈ Λ+p so that for all ζ ∈ B3 there exists an escaping sequence gn
in S such that
ρ(gn(ζ), γ)→ 0 as n→∞,
where γ is the geodesic half-ray starting at 0 and landing at z. It follows that
ρ(ζ, g−1n (γ))→ 0 as n→∞.
Now for any point w ∈ Ω− we can choose ζ to be an arbitrarily small Euclidean distance
away from w compared to the distance from w to Λ−. It follows from the geometry that
94 4. LIMIT SETS
there is some K > 0 (depending on the distance from w to Λ−) such that one of the end
points of g−1n (γ) accumulates at most a distance K(1− |ζ|) away from w. By choosing ζ
close enough to w, any point on the boundary within K(1− |ζ|) of w lies inside Ω−, and
so it must be g−1n (z) (and not g−1n (ζ)) that accumulates within K(1− |ζ|) of w. Since we
can choose 1− |ζ| to be arbitrary small, it follows that S−1(z) accumulates at w itself. As
w was arbitrary we have Ω− ⊆ S−1(z), and so Ω− ⊆ S−1(z). Since Λ+ is contained in Ω−
the result follows.
For any z ∈ D+, the closure of S−1(z) covers Λ+. The proof above shows that for z ∈ Λ+p ,
the closure of S−1(z) covers Ω−.
Let µ be the Lebesgue measure on S2, normalised so that µ(S2) = 1. For Kleinian groups,
the Lebesgue measure of the conical and point transitive limit sets are equal [34, Theo-
rem 2.4.3]. For semigroups, provided Λ+ * Λ−, the sets D+ and Λ+p either have the same
measure as Λ+c , or have measure zero. In fact this is true of any forward invariant subset
of Λ+c . At its heart, the proof given in [34, Theorem 2.4.3] uses the Lebesgue density
theorem, which can be found in [13, Equation 2.20], and we record here.
Theorem 4.7. Lebesgue density theorem For x ∈ S2 let B(x, r) be the open chordal ball
of radius r centred at x, and let µ be the Lebesgue measure on S2. If E is a measurable
subset of S2, then for µ-almost all x ∈ E we have
limr→0
µ(B(x, r) ∩ E)
µ(B(x, r))=
1 if x ∈ E
0 if x /∈ E.
Our analogous result uses hyperbolic harmonic measure, a tool familiar to Kleinian group
theorists. The hyperbolic harmonic measure of a measurable set X ⊆ S2 is the function
ωX : B3 → R
defined by
ωX(z) =
∫X
(1− |z|2
|z − ζ|2
)2
dµ(ζ).
See [34, Chapter 5] for more on hyperbolic harmonic measure.
2. ON THE STRUCTURE OF THE LIMIT SETS 95
Lemma 4.8. Let S be a semigroup and let X be any measurable forward invariant subset
of S2. If µ(Λ−c ∩X) > 0 then µ(X) = 1.
Proof. For z ∈ B3 let ωX(z) denote the hyperbolic harmonic measure of X on S2.
Then for any g ∈ S we have
ωX(0) ≥ ωg(X)(0) = ωX(g−1(0)).
The inequality follows from g(X) ⊆ X and the definition of ωX , while the equality follows
from the invariance property ωg(X)(z) = ωX(g−1(z)) for all g ∈ Aut(B3), given in [34,
Equation 5.1.3]. The conical limit of ωX(z) at a point w on the ideal boundary, is the
limit, if it exists, of ωX(z) as z converges to w from a bounded hyperbolic distance of some
geodesic landing at w. Fatou’s theorem [34, Theorem 5.1.5] tells us that the conical limit
of ωX(z) exists and agrees with the characteristic function of X at µ-almost every point
on S2. Since µ(Λ−c ∩X) > 0 there exists some w ∈ Λ−c ∩X such that ωX(z) has limit 1
as z converges to w conically. Let us choose some sequence gn ∈ S such that g−1n (0)→ w
conically. The above inequality gives
ωX(0) ≥ ωX(g−1n (0))→ 1 as n→∞.
But from the definition of hyperbolic harmonic measure, ωX(0) = µ(X). Hence µ(X) =
1.
Corollary 4.9. If S is a nonelementary semigroup such that Λ+ * Λ−, then
Λ+p ⊆ D+ ⊆ Λ+
c .
Moreover, either µ(Λ+p ) = µ(Λ+
c ) or µ(Λ+p ) = 0. Similarly for D+, either µ(D+) = µ(Λ+
c )
or µ(D+) = 0.
Proof. The inclusions follows from Theorems 4.5 and 4.6 noting that Λ+ * Λ−
implies Ω− must be nonempty. As we remarked in the introduction to this subsection,
both Λ+p and Λ+
c are Gδσ sets and so are certainly measurable, while that D+ is measurable
can be seen from its definition. Now suppose Λ+c and Λ+
p do not have the same measure,
that is µ(Λ+c ∩X) > 0 where X is the backward invariant set S2 \Λ+
p . Now by Lemma 4.8
96 4. LIMIT SETS
(applied to S−1) it follows that µ(X) = 1, and so µ(Λ+p ) = 0. The argument for D+ is
identical.
We remark that in the case where S is discrete, the argument given in the proof of
[34, Theorem 2.4.3] can be directly applied to S, and we can infer that the sets Λ+p and
Λ+c have the same Lebesgue measure.
2.3. The buried points of the limit sets. For a semigroup S we define B+(S)
to be the set of buried (sometimes called inaccessible) points in Λ+(S), that is the set of
points in Λ+(S) that do not lie on the boundary of any connected component of Ω+(S). As
usual we abbreviate B+(S) to B+ whenever the semigroup in question is clear. Because
of the topological nature of this definition, the ambient space upon which S is taken to
act is important. For example if S is the Cantor semigroup whose forward limit set is the
middle-thirds Cantor set, then, regarded as a subset of the real line, only the countably
many end points of the deleted intervals are not buried point of the limit set. However
as a subset of the complex plane, no limit point is buried. Because of the difficulties in
studying higher dimensions, we limit our study of buried points to semigroups acting on
2-dimensional hyperbolic space.
Lemma 4.10. If S is a nonelementary semigroup acting on Aut(D), then B+ is a forward
invariant, dense Gδ subset of Λ+.
Proof. Let C denote the set of components of Ω+, so that
Λ+ = S2 \⋃C∈C
C
and
B+ = S2 \⋃C∈C
C =⋂C∈C
S2 \ C =⋂C∈C
Λ+ \ C.
Hence B+ is a Gδ subset of Λ+. Each C is an open interval, the closure of which meets
the perfect set Λ+ only at its end points, and so Λ+ \C is an open, dense subset of Λ+. As
Λ+ is a Baire space it follows from the Baire category theorem that B+ is a dense subset
of Λ+. It remains to show that B+ is forward invariant. Given any z ∈ B+ and g ∈ S,
suppose towards contradiction that g(z) is not a buried point. Then g(z) ∈ ∂C for some
2. ON THE STRUCTURE OF THE LIMIT SETS 97
C ∈ C, and so z ∈ g−1(∂C). Since Ω+ is backward invariant, g−1 maps each connected
component inside another, so that g−1(∂C) ⊆ C1 for some C1 ∈ C. Since z lies in the
complement of Ω+ we must have z ∈ ∂C1, contradicting that z is a buried point. This
contradiction shows that g(z) must be a buried point after all.
Recall that D+ is also a forward invariant, dense Gδ subset of Λ+ and so B+ has some of
the same dynamical and topological properties as D+. In two dimensions, one is contained
in the other.
Proposition 4.11. If S is a nonelementary subsemigroup of Aut(D) and Λ+ 6⊆ Λ−, then
D+ ⊆ B+.
Proof. We first consider the case where Λ+ has empty interior, so that Λ+ is a Cantor
set. We can write the set of connected components of Ω+ as
(pi, qi) | i ∈ I
where I is some countable index set and (pi, qi) ⊆ S1 is the open interval contained in
the unit circle traversed by moving in an anticlockwise direction from pi to qi. We write
P = pi | i ∈ I and Q = qi | i ∈ I, so that Λ+ \B+ = P ∪Q. Notice that since Λ+
has no isolated points, P and Q are disjoint. Suppose there exists some z that lies in both
Λ+ \B+ and D+. Further suppose z ∈ P and write z = pj ; we shall show that Λ− covers
Q. Since z ∈ D+, S−1(z) accumulates at every point in Λ+ and in particular at every
point in Q. This means that for any q ∈ Q there exists a sequence gn ∈ S such that
limn→∞
g−1n (pj) = q.
Recall that g−1 is an orientation-preserving self-map of S1 that maps components of Ω+
within components of Ω+. Hence there is some sequence k(n) in I such that
g−1n (pj , qj) ⊆ (pk(n), qk(n)).
One possibility is that there exists some subsequence of k(n) whose elements are distinct.
In this case we can assume without loss of generality that each k(n) is distinct. Hence the
diameters of (pk(n), qk(n)) tend to zero as n tends to infinity, and since g−1n (pj) converges
98 4. LIMIT SETS
to q, so does g−1n (ζ) for every ζ ∈ (pj , qj). Hence g−1n converges ideally to q, and so q ∈ Λ−.
Otherwise there is some constant subsequence of k(n), and in this case, without loss of
generality, we can assume the sequence itself is constant. Hence
g−1n (pj , qj) ⊆ (pk, qk)
for some k ∈ I and all n. Since g−1n (pj) converges to a point in Q and P ∩Q = ∅, we must
have q = qk. Again using that g−1n preserves orientation, g−1n (ζ) converges to qk for every
point ζ in (pj , qj). This means that g−1n converges ideally to qk, and so q = qk ∈ Λ−. By
symmetry of argument if z lies in Q and D+, then Λ− covers P . Hence the existence of
any point z in both Λ+\B+ and D+ implies that Λ− covers at least one of the sets P or Q.
We claim that both P and Q are dense in Λ+. Then if Λ+\B+ and D+ meet, then P ⊆ Λ−
or Q ⊆ Λ−; and so by taking closures we obtain Λ+ ⊆ Λ−, contrary to hypothesis. It
follows from this contradiction that no such z exists and so Λ+ \B+ ⊆ Λ+ \D+, that is,
D+ ⊆ B+ as required.
It now suffices to prove the claim. Each point in P is the limit of points in Q, that is
P ⊆ Q, and so P ⊆ Q. Similarly Q ⊆ P and so Q = P , and it follows that
Λ+ = P ∪Q = P ∪Q = P = Q.
This completes the proof of the claim and proves the result in the case where Λ+ has
empty interior.
Now suppose Λ+ has nonempty interior. If Λ+ is equal to S1, then B+ is also equal to
S1; hence D+ ⊆ B+. Otherwise we can choose a point z in Λ+ \B+. In this case Ω+ is a
backward invariant set that does not cover Λ+. Since z lies in Λ+ it lies in the closure of
Ω+, but by the definition of B+ it cannot lie in the interior of Ω+, and so z ∈ ∂Ω+. Since
Ω+ is backward invariant it follows that S−1(z) does not meet the complement of Ω+, that
is, the interior of Λ+. Hence z cannot lie in D+, which shows that Λ+ \ B+ ⊆ Λ+ \D+;
in other words D+ ⊆ B+, as required.
2. ON THE STRUCTURE OF THE LIMIT SETS 99
Notice that in the case where Λ+ has nonempty interior, we did not use the fact that
S ⊆ Aut(D). This means that in all dimensions D+ ⊆ B+ whenever Λ+ has a nonempty
interior.
We ask whether or not the reverse inclusion of Proposition 4.11 holds, that is, do we have
B+ ⊆ D+ under suitable hypotheses? The answer to this question is no, as can be seen
by describing D+ and B+ explicitly in the case of a very well behaved and tractable semi-
group, the Cantor semigroup, which we recall is generated by the maps fi(z) = 13(z + i)
for i = 0, 2. We remark that although the Cantor semigroup is an elementary semigroup,
the following descriptions of D+ and B+ can be seen to generalise; for example to sub-
semigroups of Aut(D) that have disjoint limit sets and are generated by a finite set F ,
with the property that f(Λ+) ∩ g(Λ+) = ∅ for all distinct f, g ∈ F . This means that the
reverse inclusion of Proposition 4.11 fails for a large class of finitely-generated nonelemen-
tary semigroups.
Let S = 〈f0, f2〉, the Cantor semigroup. Each point z in Λ+ has a unique base–3 expansion
z = 0.i1i2 . . . in . . .
where each in lies in 0, 2. We consider words, both infinite and finite, whose letters lie in
0, 2. It is clear that B+ is the set of those points z = 0.i1i2 . . . in . . . in Λ+ such that in
is not eventually equal to 0 or eventually equal to 2. Since the forward and backward limit
sets of S are disjoint we have Λ+c = Λ+. On the other hand, D+ is the set of points in Λ+
whose base–3 expansion contains every finite string in the letters 0, 2 as a sub-string.
To see this, first suppose i1i2 . . . in . . . is an infinite word in 0, 2 with this property.
Then for any finite word w there are natural numbers m < n such that w = im . . . in.
Hence f−1im−1· · · f−1i1 (z) is a point in Λ+ whose base–3 expansion begins with w. Since w
was arbitrary, S−1(z) accumulates at every point in Λ+, and so z ∈ D+ by definition.
Conversely suppose z ∈ Λ+ has base–3 expansion 0.i1i2 . . . in . . .. Suppose that some finite
word, w = w1 . . . wm say, does not feature anywhere in the infinite word i1i2 . . . in . . .. We
claim that S−1(z) does not meet fw1 · · · fwm([0, 1]). The claim follows since for any infinite
100 4. LIMIT SETS
word j1j2 . . . jn . . . and any p ∈ N we have
0.j1j2 . . . jn . . . ∈ fj1 · · · fjp([0, 1]),
and that f−1jp · · · f−1j1
(0.i1i2 . . . in . . .) ∈ (0, 1) if and only if jq = iq for every q = 1, 2, . . . , p.
Hence by the claim, S−1(z) avoids the interior of fw1 · · · fwm([0, 1]) and so is not dense in
Λ+. This implies that z cannot lie in D+.
It is interesting to consider how the set of attracting fixed points fits in this picture. It
turns out that every attracting fixed point of S, except those of the generators f0 and
f2, lies in B+, but no attracting fixed points lie in D+. Indeed, if i ∈ 0, 2n then the
attracting fixed point of Fi = fi1 · · · fin has base–3 expansion 0.i1 . . . ini1 . . . ini1 . . . in . . ..
This is clear upon noting that αFi ∈ Fi([0, 1]), and αFki= αFi for each positive integer k.
Hence
αFi ∈ F ki ([0, 1]) =
k copies of fi1 ...fin︷ ︸︸ ︷fi1 . . . finfi1 . . . fin . . . . . . fi1 . . . fin([0, 1])
for all k. Recall that a point lies in B+ exactly when its address is not eventually constant.
Hence for every n > 1 and every i ∈ 0, 2n, the point αFi is a buried point. Any nonele-
mentary subsemigroup of Aut(D) contains many subsemigroups for which the conclusions
above are valid, and so many of its attracting fixed points are also buried points in Λ+.
In fact our description of D+ generalises beyond subsemigroups of Aut(D): Suppose S
is generated by F = f1, . . . , fm, where each fj lies in Aut(B3). Further suppose that
the limit sets of S are disjoint. Then, using ideas similar to those used above, it is pos-
sible to show that D+ is the collection of points in Λ+ for which there exists a sequence
in ∈ 1, 2, . . . ,m for each n ∈ N, where every finite sequence taking values in 1, 2, . . . ,m
occurs somewhere along the infinite sequence (i1, i2, . . . , in, . . .).
Returning to the Cantor semigroup, we show that its forward point transitive limit set,
Λ+p , is empty. To see this, recall that, as remarked after the proof of Theorem 4.6, for
any z ∈ Λ+p the closure of S−1(z) covers Ω−. Accordingly we claim that for any z ∈ Λ+
the orbit S−1(z) is not dense in Ω− = R. Choose z ∈ Λ+ with base–3 expansion 0.i0i1 . . .
2. ON THE STRUCTURE OF THE LIMIT SETS 101
and choose any j = (j0, . . . , jn−1) ∈ 0, 2n. If Fj = fj0 · · · fjn−1 then F−1j (z) ∈ [0, 1] if
and only if (i0, . . . , in−1) = (j0, . . . , jn−1). Moreover if (i0, . . . , in−1) = (j0, . . . , jn−1), then
F−1j (z) ∈ Λ+. Hence S−1(z) does not meet [0, 1] \ Λ+, which proves the claim.
It should perhaps not be surprising that the Cantor semigroup has an empty point tran-
sitive limit set, as it is known (see [34, Theorem 2.3.3]) that a Kleinian group whose limit
set is not the full ideal boundary necessarily has an empty point transitive limit set. We
do not know if something similar is true for semidiscrete semigroups.
2.4. The limit sets of the group and inverse free parts of a semigroup. Recall
from Section 1 that the inverse free part of a semigroup S is the semigroup S \ S−1 of
elements in S whose inverses do not belong to S. In this section we consider the limit sets
of S \ S−1 when S is not a group. Throughout we let G denote S ∩ S−1, the group part
of S.
Theorem 4.12. If S is a semigroup and is not a group, then Λ+(S) = Λ+(S \ G) and
Λ−(S) = Λ−(S \G). Moreover we have Λ+c (S) = Λ+
c (S \G) and Λ−c (S) = Λ−c (S \G).
Proof. Clearly Λ+(S \ G) ⊆ Λ+(S). We show that Λ+(S) ⊆ Λ+(S \ G). For any
z ∈ Λ+(S), choose a sequence gn ∈ S such that
limn→∞
gn(0) = z.
Now choose h ∈ S \G and note that
limn→∞
gnh(0) = z.
Lemma 2.1 tell us that gnh lies in S \G, so that we have z ∈ Λ+(S \G). Hence we have
shown that
Λ+(S \G) = Λ+(S).
Moreover, if gn(0) converges to z conically, then gnh(0) also converges to z conically, and
so this reasoning also tells us that Λ+c (S) = Λ+
c (S \ G). As usual, the corresponding
statements for the backward limit sets follow upon replacing S with S−1 in the preceding
arguments.
102 4. LIMIT SETS
The theorem above tells us that provided a semigroup S is not a group, the limit sets and
conical limit sets of the inverse free part of S are equal to the corresponding limit sets of
S itself. In fact the same is true of both horospherical limit sets. In order to prove this
we need a geometrical lemma. Recall that [α, β] denotes the hyperbolic geodesic between
two points α and β in hyperbolic space. Similarly (α, β] denotes the hyperbolic geodesic
between α and β with the point α removed. Throughout the remainder of this section,
for any set X ⊆ B3 and M ≥ 0 we let XM denote the set of points that are less than a
hyperbolic distance of M from some point in X.
Lemma 4.13. Given any x ∈ B3 and y ∈ (z, x] we have
(Hz(y))ρ(x,y) = Hz(x).
Proof. It is convenient to work in the upper half-space model and conjugate so that
z = ∞. Then Hz(y) =w ∈ H3 | ht[w] > ht[y]
for any y ∈ H3. For any t > 0 the
set of points that are less than a hyperbolic distance t from Hz(y) is exactly the horoball
(Hz(y))t =w ∈ H3 | ht[w] > ht[y − ju]
where u > 0 is such that t = ρ(y − ju, y).
It follows that for any y ∈ [x,∞) we have (Hz(y))ρ(x,y) =w ∈ H3 | ht[w] > ht[x]
=
Hz(x).
Theorem 4.14. If S is a semigroup and is not a group, then
Λ+h (S \G) = Λ+
h (S).
Proof. Let Λ+h,x(S) denote the points z ∈ Λ+(S) such that some horoball based at z
does not meet the S-orbit of x. It is easy to see that
Λ+h,x(S) = Λ+
h (S)
for all x ∈ B3. Using Lemma 4.13 we show that
Λ+h,x(S \G) = Λ+
h,x(S)
for any x ∈ B3.
Given any x ∈ B3 we certainly have
Λ+h,x(S \G) ⊆ Λ+
h,x(S).
2. ON THE STRUCTURE OF THE LIMIT SETS 103
To show the reverse inclusion suppose z is a point in Λ+h,x(S) and gn is a sequence in S
such that the sequence of points gn(x) meets any horoball based at z. By assumption we
can choose some f /∈ G, and again appealing to Lemma 2.1, we see that the element gnf
lies in S \G, itself a semigroup. Moreover
ρ(gnf(x), gn(x)) = ρ(f(x), x).
Let Hz(u) be any horoball based at z. Then by the claim we can choose v ∈ B3 such that
(Hz(v))ρ(f(0),0) ⊆ Hz(u).
Since gn(x) meets Hz(v) it follows gnf(x) meets Hz(u). Since Hz(u) was arbitrary we
have a sequence of points gnf(x) in the S \G orbit of x which meets every horoball based
at z. Hence we have shown that
Λ+h,x(S \G) = Λ+
h,x(S),
and so
Λ+h (S \G) = Λ+
h (S).
Theorem 4.15. Either Λ+(S) = Λ(G) or Λ(G) ⊆ Λ+ \ Λ(G). In other words Λ+(S) =
Λ(G) or Λ+ = Λ+ \ Λ(G).
Proof. Each g ∈ G fixes Λ setwise, and so each g maps Λ+(S) \ Λ(G), and hence
its closure, into itself. Now either Λ+(S) \ Λ(G) has less than two points, in which case
Λ+(S) = Λ(G), or as the smallest G-forward invariant set containing at least two points,
Λ+(G) is contained in Λ+(S) \ Λ(G). But Λ+(G) = Λ(G) which gives the result.
As usual the analogous version of the theorem above for the backward limit set follows
upon replacing S with S−1. Shown in Figure 4.3 are the limit sets of the semidiscrete
semigroup found by taking the Kleinian group G generated by (see [33, page 272]) the
two transformations
z 7→ (0.8210− 0.3832i)z + (0.3832 + 1.8210i)
(−0.3832 + 0.1789i)z + (0.8210− 0.3832)
104 4. LIMIT SETS
Figure 4.3. Limit sets of a semigroup with nonempty group and inverse
free parts
and
z 7→ (1− i)z + 1
z + (1 + i),
then, in the spirit of Theorem 2.20, appending the generator
z 7→ (2.77− 1.63i)z + (2.25− 5.22i)
(1.74 + 0.0213i)z + (2.77− 1.63i).
The forward limit set is in red, while the backward limit set is in blue. They intersect on
the purple set, which is the limit set of the Kleinian group G. Each point in the purple
set is a limit point of the red set, and a limit point of the blue set.
3. ON SEMIGROUPS WITH DISJOINT LIMIT SETS 105
In general, if a semidiscrete semigroup S has a nonelementary group part G, then Λ+(S)∩
Λ−(S) contains Λ(G), necessarily a perfect set. It is an open problem, even in two di-
mensions, whether or not the converse is true. That is, if S is a semidiscrete semigroup
and Λ+(S)∩Λ−(S) is a perfect set, must S necessarily have a nonelementary group part?
This question is particularly interesting in view of Theorems 3.33 and 3.35.
3. On semigroups with disjoint limit sets
In this section we study semidiscrete and inverse free semigroups that are finitely-generated
and have disjoint limit sets. We shall show in Theorem 4.16 that if a finitely-generated
semigroup has disjoint limit sets, then, provided it has no elliptic generators, it is semidis-
crete and inverse free too. Semigroups that belong to this class are very well behaved,
even in higher dimensions. For example, in [15] the authors show that given a finitely-
generated semigroup S acting in three dimensions and with disjoint limit sets, there exists
a metric (which they construct) defined on some open set containing Λ+, with respect to
which the semigroup can be regarded as a contracting iterated function system with limit
set Λ+. We shall show that if we further suppose S is semidiscrete and inverse free, then
there exists a finitely-generated subsemigroup of S that can be regarded as a contracting
iterated function system with respect to the chordal metric on some open set containing
Λ+, and which has limit set Λ+.
In Chapter 3 we characterised semidiscrete and inverse free semigroups in two dimensions.
In higher dimensions we have the following.
Theorem 4.16. Suppose F is a finite subset of Aut(B3) such that the identity and any
elliptic maps in F generate a group of finite order. If the limit sets of S = 〈F〉 are disjoint,
then S is semidiscrete. If moreover F does not contain the identity or any elliptic maps,
then S is inverse free.
Proof. Since the limit sets of S are disjoint, F contains no parabolic elements. The
identity and any elliptic elements in F fix Λ+ setwise, as their inverses also lie in S and
Λ+ is forward invariant. Any other elements in F are loxodromic. If any loxodromic
f ∈ F were to fix Λ+, then both fixed points of f must lie in Λ+. This is not possible as
106 4. LIMIT SETS
the repelling fixed point of f lies in Λ−, which by assumption does not meet Λ+. Hence
each loxodromic element in F maps Λ+ strictly inside itself. It now follows that S is
semidiscrete by Theorem 2.5. If moreover F does not contain the identity or any elliptic
maps, then F must be purely loxodromic and no element in F fixes Λ+ setwise. Hence,
again by Theorem 2.5, S is also inverse free.
For any f ∈ Aut(B3) we define the quantity L(f) by
L(f) = supz,w∈S2z 6=w
χ(f(z), f(w))
χ(z, w),
and shall make use of the following equation given in [3, Theorem 3.6.1]
(12) L(f) = exp[ρ(0, f(0))].
We shall make repeated use of the following important property enjoyed by all semigroups
generated by a bounded set. Recall that F denotes the closure of F in Aut(B3).
Lemma 4.17. Let S be a semigroup generated by some bounded set F . Then
Λ+(S) =⋃f∈F
f(Λ+(S)).
Proof. One inclusion is clear from the forward invariance of Λ+ = Λ+(S). For the
other inclusion, suppose z ∈ Λ+ and fix some sequence gn ∈ S such that gn(0) → z. We
show z lies in f(Λ+) for some f ∈ F . For each n we can choose fn ∈ F and hn ∈ S ∪ I
such that gn = fnhn. By compactness we can pass to a subsequence, without loss of
generality the sequence itself, such that fn → f ∈ F . Now it follows that χ(fnhn(0), z)→
0 as n→∞, and so
1
L(f−1n )χ(hn(0), f−1n (z))→ 0 as n→∞,
by the definition of L(f−1n ). Making use of equation (12), for each f ∈ Aut(B3) we have
L(f) = exp[ρ(0, f(0))] = exp[ρ(f−1(0),0)] = L(f−1), and as F is bounded, it follows
that L(f) | f ∈ F is bounded above. Hence χ(hn(0), f−1(z)) → 0 as n → ∞, and so
f−1(z) ∈ Λ+. We have shown that for all z ∈ Λ+ there exists f ∈ F such that z ∈ f(Λ+).
Hence we have proven the other inclusion.
3. ON SEMIGROUPS WITH DISJOINT LIMIT SETS 107
Theorem 4.18. Let S be a semigroup generated by a finite set F . If S is semidiscrete
and inverse free and the limit sets of S are disjoint, then for each point p in Λ+ there
is a composition sequence generated by F that converges conically to p. Moreover, every
composition sequence converges conically to a single point on the ideal boundary, and in
particular, converges ideally.
Proof. We first show that each point z in Λ+ also lies in Λ+c . By Lemma 4.17 there
exists f1 ∈ F and z1 ∈ Λ+ such that z = z0 = f1(z1). Similarly there exists f2 ∈ F and
z2 ∈ Λ+ such that z1 = f2(z2) so that z = f1f2(z2). Continuing in this way we recursively
choose fn ∈ F and zn ∈ Λ+ such that zn−1 = fn(zn). Writing Fn = f1 · · · fn we have
z = Fn(zn) so that
zn = F−1n (z).
Since I /∈ S the sequence Fn is escaping. By Theorem 1.2 if z did not lie in the backward
conical limit set of the sequence F−1n , then zn must accumulate only in Λ−; but this cannot
be as zn is a sequence in Λ+. Hence we must have
z ∈ Λ−c (F−1n ) = Λ+c (Fn) ⊆ Λ+
c (S)
as claimed. In fact Fn converges to z ideally. Indeed, we show that every composition
sequence converges ideally, as the theorem also asserts. To see this, consider C(Λ+), the
convex hull of Λ+, that is the smallest hyperbolically convex subset of B3 containing every
geodesic whose landing points both belongs to Λ+. As Λ+ is forward invariant under S,
so is C(Λ+). Since the limit sets are disjoint and F−1n is escaping, ρ(F−1n (0), C(Λ+))→∞
as n→∞. Hence ρ(0, Fn(C(Λ+))) = ρ(0, C(Fn(Λ+)))→∞ as n→∞. Since Fn(Λ+) is a
decreasing nested sequence of sets and its chordal diameter converges to 0, z is the only
accumulation point of Fn. It follows that Fn(0) converges conically to z.
Recall that for a semigroup S, the set Λ+q is the collection of points in Λ+ that are limit
points of some composition sequence generated by S. Hence for a semigroup S as above,
Λ+ = Λ+c = Λ+
q . Furthermore, since S has disjoint limit sets and is semidiscrete and inverse
free, then S−1 also enjoys these properties. It follows that we also have Λ− = Λ−c = Λ−q .
We cannot drop the assumption that F is bounded, as the following example demonstrates.
108 4. LIMIT SETS
Take loxodromic elements g and h in Aut(D) whose axes are configured as in Figure 4.4
below.
hg
Figure 4.4. Loxodromic maps g and h
Let X be the complementary component of S1 \ αg, αh that does not meet either gen-
erator’s repelling fixed point. Let F be the unbounded setgh, g2h2, g3h3, . . .
and
let S be the semigroup generated by F . Then αg ∈ Λ+(S) since gnhn(0) → αg as
n → ∞. Yet for any composition sequence Fn generated by F , we have, for all n,
Fn(X) = f1 · · · fn(X) ⊆ f1(X) = gmhm(X) for some m ∈ N. Hence Fn(0) can only
accumulate in gmhm(X), and in particular, Fn(0) does not accumulate at αg. This means
that Λ+q (S) is not equal to Λ+(S) for this particular example.
3.1. A sufficient condition for discreteness. In [10, Theorem 10] the author
shows that if a finitely-generated subsemigroup of Aut(D) has disjoint limit sets, then
the semigroup must be discrete. In this subsection we show that the same is true in all
dimensions if we additionally suppose S is generated by a set of loxodromic elements.
Recall that if X is a subset of a topological space, its derived set, denoted X ′, is the set of
limit points of X. Hence a semigroup S is discrete precisely when S′ is empty. It is easy
to check that S is also a semigroup whose limit sets coincide with those of S. Motivated
by this, we begin by studying S′, which it turns out is a semigroup in its own right. For
any A,B ⊆ Aut(B3) we define AB to be the set ab | a ∈ A and b ∈ B.
Theorem 4.19. Suppose S is a semigroup, then
(i) S′ is a semigroup,
(ii) SS′ and S′S are subsets of S′, and
3. ON SEMIGROUPS WITH DISJOINT LIMIT SETS 109
(iii) if S′ is nonempty then Λ+(S) = Λ+(S′) and Λ+c (S) = Λ+
c (S′). Similarly Λ−(S) =
Λ−(S′) and Λ−c (S) = Λ−c (S′).
Proof. We must show S′ is closed under composition. Suppose that g, h ∈ S′, say
g = limn→∞
gn
and
h = limn→∞
hn
where gn and hn are sequences of distinct elements in S \ g and S \ h respectively.
Since the group of Mobius transformations is a topological group we have
gh = limn→∞
gnhn.
If gnhn contains a subsequence of distinct elements in S \ gh then gh ∈ S′. Otherwise
gnhn = gh for all large enough n. Since hn 6= hn+1 we must have gnhn+1 6= gh, and as
gh = limn→∞
gnhn+1
we have gh ∈ S′ in this case too.
To see SS′ ⊆ S′ suppose again that g ∈ S′ via
g = limn→∞
gn
where gn are distinct elements in S \g. Now for any h ∈ S, hgn is a sequence of distinct
points in S \ hg and converges to hg. Hence hg lies in S′ as claimed.
Now suppose S has a nonempty derived set. We first show that Λ+(S) ⊆ Λ+(S′). Working
in the ball model, suppose z ∈ Λ+(S) and choose a sequence fn in S such that
limn→∞
fn(0) = z.
Since S′ is nonempty we can choose some g ∈ S′. Since
limn→∞
fng(0) = z
110 4. LIMIT SETS
and SS′ ⊆ S′, it follows that z ∈ Λ+(S′).
For the converse, suppose z lies in Λ+(S′) and hn is a sequence in S′ such that
limn→∞
hn(0) = z.
Since each hn lies in S′ we can choose a sequence gn in S such that σ(gn, hn) < 1/n for
each n ∈ N. Hence χ(gn(0), hn(0)) < 1/n, so that gn(0) also converges to z, and we see
that z ∈ Λ+.
To show the conical limit sets of S and S′ coincide, we suppose that fn is a sequence in
S and that fn(0) → z conically as n → ∞. Since ρ(fng(0), fn(0)) = ρ(g(0),0) it follows
that for any g ∈ S′ fng(0) converges to z conically. Since SS′ is contained in S′ we see
that Λ+c (S) ⊆ Λ+
c (S′). For the reverse inclusion we use a version of the argument we used
to show Λ+(S) ⊆ Λ+(S′): Suppose z lies in Λ+c (S′) via some sequence hn in S′ such that
hn(0) converges to z conically. Since the map that sends any g ∈ Aut(B3) to g(0) ∈ B3
is continuous, it follows that we can choose gn ∈ S such that ρ(gn(0), hn(0)) < 1. Hence
gn(0) converges to z conically, and so z lies in Λ+c (S).
As usual the results for the backward limit sets follow upon considering S−1 instead of S,
and noting that the map g → g−1 from Aut(B3) to itself is a homeomorphism.
Before proving the main result of this subsection, we need a preparatory lemma.
Lemma 4.20. Suppose S is a semigroup generated by a finite set F . Then for all g ∈ S′
there exists f ∈ F such that f−1g ∈ S′.
Proof. Given any g ∈ S′ we can choose a sequence of distinct elements gn ∈ S \ g,
such that gn → g. Since F is finite, for some f ∈ F we can pass to a subsequence, which
we do not rename, such that
gn = fhn,
where hn is a sequence of distinct elements in ∈ S \f−1g
. Since hn → f−1g it follows
that f−1g ∈ S′ as claimed.
3. ON SEMIGROUPS WITH DISJOINT LIMIT SETS 111
Theorem 4.21. Let S be a semigroup generated by a finite set F . If I /∈ S and the limit
sets Λ+ and Λ− are disjoint, then S is discrete.
Proof. We show that if S is not discrete then Λ+ and Λ− must meet. If S is not
discrete then S′ is nonempty so we can choose g ∈ S′. Repeatedly applying Lemma 4.20
above, we can generate a sequence fn ∈ F such that
f−1n · · · f−11 g ∈ S′
for each n. We let Fn = f1 · · · fn for each n ∈ N. Since I /∈ S, the composition sequence
Fn is escaping by Theorem 2.7; hence both F−1n g and F−1n are escaping sequences, and in
particular have at least one accumulation point on S2. The sequence F−1n g lies in S′ while
F−1n lies in S−1. Since
ρ(F−1n g(0), F−1n (0)) = ρ(g(0), (0))
any accumulation point ζ of F−1n (0) is an accumulation point of F−1n g(0), so that ζ lies
in both Λ−(S) and Λ+(S′). Finally by Proposition 4.19 Λ+(S′) = Λ+(S), and so Λ+(S)
meets Λ−(S) at ζ.
3.2. Perturbed semigroups. In the following subsection we investigate what hap-
pens to the limit sets of a semigroup S = 〈F〉 when its generating set F , considered as an
|F|-tuple of Mobius transformations, is perturbed. By perturbing a semigroup we mean
perturbing each of its generators in Aut(B3). Most of the results given here assume that
F is finite, although we occasionally only require F to be bounded.
Suppose that S is a finitely-generated, semidiscrete and inverse free semigroup. Our first
theorem indicates that if the limit sets of S are disjoint, then they behave nicely as the
generators of S are perturbed. Before stating this theorem more precisely, we first give an
important lemma, which we shall use again at the close of this chapter.
Lemma 4.22. Let S be a semidiscrete and inverse free semigroup generated by a finite set
F . Then for all M > 0 there exists a finite subset FM of S, such that for all f ∈ FM ,
ρ(f(0),0) > M , and Λ+(S) = Λ+(SM ) where SM is the semigroup generated by FM . In
112 4. LIMIT SETS
particular,
Λ+(S) =⋃
f∈FM
f(Λ+(S)).
Proof. Choose M > 0 and let us write N = 0, 1, . . . , N − 1, so that Nn is the set
of n tuples on the set 0, 1, . . . , N − 1. Let T = N<ω be the union⋃∞n=1N
n, which we
endow with the usual partial order relation, that is, for s, t ∈ T , then s ≤ t if and only if
the domain of s is contained in the domain of t, and s agrees with t on the domain of s.
Equipped with this partial ordering T is a tree. Letting F = f0, . . . , fN−1, the elements
of T may be regarded as words in F , and so as elements of S. More precisely we make
use of the projection
π : T → S
given by
π((i0, . . . , iN−1)) = fi0 · · · fiN−1 .
The projection is injective exactly when S is free. The nth level of T is the set Nn, which
corresponds to the collection of n-length words generated from F . The level to which
an element in T belongs is called its height, and we write ht(t) to denote the height of
t ∈ T . We say two elements s, t ∈ T are comparable if s ≤ t or t ≤ s, otherwise they are
incomparable. A cofinal branch of T is any maximal, linearly ordered subset of T that
meets every level.
To prove the lemma we find a set TM ⊆ T such that;
(i) TM is a finite set of pairwise incomparable elements,
(ii) ρ(π(t)(0),0) > M for each t ∈ TM ,
(iii) for each t ∈ T there is some (unique by (i), although we do not use uniqueness)
s ∈ TM such that s and t are comparable, and,
(iv)
Λ+ =⋃t∈TM
π(t)(Λ+).
In order to find TM , we recursively construct a monotonically increasing nest of sets
TM (n) ⊆ T . We define TM (0) = ∅. Suppose that for some n ∈ N the set TM (n − 1) has
3. ON SEMIGROUPS WITH DISJOINT LIMIT SETS 113
been constructed. Now consider all elements in the nth level of T which are not greater
than some element in TM (n− 1), that is the set
Ln = Nn \ t ∈ T | s ≤ t for some s ∈ TM (n− 1) .
Now put
TM (n) = TM (n− 1) ∪ t ∈ Ln | ρ(π(t)(0),0) > M .
We show that Ln is empty for some n. For if not, then
T ′ =⋃n∈N
Ln
is a tree of height ω. We claim T ′ is a tree. To see this, it is enough to show that for
any t ∈ T ′ and s ≤ t we have s ∈ T ′. Suppose towards contradiction that s /∈ T ′. Then
s /∈ Lht(s), so that r ≤ s for some r ∈ TM (ht(s)− 1) ⊆ TM (ht(t)− 1). But then t ≤ s ≤ t
which contradicts t ∈ Lht(t). Hence T ′ is a tree as claimed. Since T ′ has height ω and
each level is finite, Konig’s lemma (see [23, Lemma 14.2]) implies it contains a linearly
ordered subset of T that intersects every level, j ∈ Nω say. The associated sequence of
Mobius transformations has nth term Fn = fj0 · · · fjn−1 . This is a composition sequence
satisfying ρ(Fn(0),0) ≤ M for all n, contradicting our assumption that S is semidiscrete
and inverse free by Theorem 2.7. Hence each Ln is eventually empty after all. This means
that there is a set, which we take to be TM , such that TM (n) = TM for all large enough
n. We set FM = π(TM ), and verify that TM has all four properties claimed. To see (i),
since each Ln is eventually empty, TM is finite. For each n when constructing the set
TM (n) we only append elements to TM (n − 1) that are incomparable with each other,
and with every element already in TM (n − 1). It follows that any two elements in TM
are incomparable, giving (i). Property (ii) clearly holds by the construction. If (iii) were
false, then for some t ∈ T there would be no s ∈ TM such that s < t. But then Ln is
not empty for each n ≥ ht(t), a contradiction. To see (iv), first let h be the maximum of
the heights of elements in TM , and choose any z ∈ Λ+. Then there is a sequence gn ∈ S
such that gn(0) → z as n → ∞. Now by property (iii) any t ∈ T may be written as
the concatenation of elements in TM , followed by some element of height no more than h.
114 4. LIMIT SETS
Since each Mobius map gn lifts to an element in T , each gn can be expressed as
gn = fi(1,n)fi(2,n) · · · fi(l(n),n)rn,
where each fi(j,n) belongs to TM , i(m,n) ∈ N and rn ∈ S where rn, regarded as a word in
F , has length no more than h. Since F is finite, the set of compositions of no more than
h elements from F forms a bounded set, hence
limn→∞
fi(1,n)fi(2,n) · · · fi(l(n),n)(0) = z.
This shows that Λ+(S) ⊆ Λ+(SM ), and as the reverse inclusion is clear we have Λ+(S) =
Λ+(SM ). Since FM is finite we can apply Lemma 4.17 to the semigroup SM and infer
(iv).
This lemma verifies the statement made in the introduction to this section, that a finitely-
generated, semidiscrete and inverse free semigroup with disjoint limits sets can be regarded
as a contracting iterated function system with respect to the chordal metric on some open
neighbourhood of Λ+, and with limit set Λ+. To see this, first recall from Chapter 1 that
for a loxodromic map f , the isometric disc of f , which we shall denote by I+(f), is the
open chordal disc contained in C where the chordal derivative of f is greater than 1. We
let I−(f) denote the isometric disc of f−1, and note that I−(f), contains αf while I+(f)
contains βf . Both isometric discs have radius 1/ sinh[12ρ(f(0),0)]. The action of f is to
map the interior of the complement of I+(f) onto I−(f). If S = 〈F〉 is a finitely-generated,
semidiscrete and inverse free semigroup with disjoint limits sets, then we can choose M
large enough such that the isometric discs of each map in FM do not meet Λ+. Then
SM = 〈FM 〉 is a subsemigroup of S and inherits its limit sets. Moreover each f ∈ FMis a strict contraction (with respect to the chordal metric) on Λ+. This shows that given
a finitely-generated semidiscrete and inverse free semigroup whose limit sets are disjoint
and some positive real number M , we can always find a subsemigroup whose generators
all have norm at least M and whose limit sets are equal to those of the original semigroup.
So far in this thesis, we have regarded the generating set of a semigroup as an unordered
set of Mobius transformations. In this subsection it is sometimes convenient to enumerate
F as a (finite) sequence, and instead regard the generators of a semigroup as a point in
3. ON SEMIGROUPS WITH DISJOINT LIMIT SETS 115
Aut(B3)|F|. To emphasise this new point of view, instead of referring to F ⊆ Aut(B3) as
a generating set, we shall say F is a generating sequence. We shall say that the generating
sequence F is bounded if the range of F is bounded in Aut(B3), and we shall denote by
〈F〉 the semigroup generated by the range of F . For k ∈ N we endow Aut(B3)k with
a product metric: we define the distance between points (f1, . . . , fk) and (g1, . . . , gk) in
Aut(B3)k by maxi=1,...,k σ(fi, gi). In fact the choice of metric is not particularly important
and we shall not need its explicit form again. Indeed, any of the usual product metrics
which induce the same topology as the one above will suffice for our purposes.
Theorem 4.23. Suppose S is generated by the finite set f1, . . . fk ⊆ Aut(B3) and that
S is semidiscrete and inverse free. If the limit sets of S are disjoint, then in some neigh-
bourhood of F = (f1, . . . , fk) ∈ Aut(B3)k (with respect to the product topology on Aut(B3))
the functions from Aut(B3)k to the collection of compact subsets of S2 (endowed with the
usual Hausdorff metric) given by
F 7→ Λ+(〈F〉) and F 7→ Λ−(〈F〉)
are continuous.
Proof. By symmetry of argument we only need to show that the function F 7→
Λ+(〈F〉) is continuous. We show that for all small enough ε > 0 there exists δ > 0 such
that whenever G is within δ of F in Aut(B3)k, we have Λ+(〈G〉) within ε of Λ+(〈F〉) in
the Hausdorff metric. Let πF : k<ω → 〈F〉 denote the projection map that sends an
element in the tree k<ω to its associated Mobius transformation in 〈F〉. By Lemma 4.22
there exists T ⊆ k<ω such that Λ+(〈πF (T )〉) = Λ+(〈F〉) and such that for all t ∈ T , the
isometric disc of πF (t) has diameter smaller than ε/6. We shall assume that the distance
between Λ+(〈F〉) and Λ−(〈F〉) is larger than 3ε, so that for no s, t ∈ T does I−(πF (s))
meet I+(πF (t)). As f is contracting with respect to the chordal metric in the interior of
S2 \ I+(f), the set F is a contracting iterated function system on
S2 \⋃f∈F
I+(f)
116 4. LIMIT SETS
(with respect to the chordal metric) and has limit set Λ+(〈F〉), which is contained in⋃f∈F
I−(f).
Each point in Λ+(〈F〉) is certainly within ε/6 of some point in⋃f∈F I
−(f). Conversely,
given any f ∈ F and any point in z ∈ I−(f), the point αf , which lies in Λ+(〈F〉), is within
ε/6 of z. Hence z lies in the ε/6 neighbourhood of Λ+(〈F〉) and it follows that
(13) dH
( ⋃t∈T
I−(πF (t)), Λ+(〈F〉)
)< ε/6.
The choice of T ensures that for any G ∈ Aut(B3)k we have Λ+(〈πG(T )〉) = Λ+(〈G〉). We
choose an open neighbourhood U of F small enough such that for each G ∈ U and t ∈ T
the disc I−(πG(t)) is within ε/3 of I−(πF (t)) and I+(πG(t)) is within ε/3 of I+(πF (t))
(both with respect to the Hausdorff metric).
Hence we have
(14) dH
( ⋃t∈T
I−(πF (t)),⋃t∈T
I−(πG(t))
)< ε/3.
Moreover, we can ensure U is small enough such that the diameter of each I−(πG(t)) is
less than ε/3, and so by the same argument used to derive (13) we see that
(15) dH
( ⋃t∈T
I−(πG(t)), Λ+(〈G〉)
)< ε/3.
Hence by the triangle inequality the relations (13), (14) and (15) imply that
dH
(Λ+(〈F〉), Λ+(〈G〉)
)< ε.
This shows that F 7→ Λ+(〈F〉) is continuous. By symmetry of argument F 7→ Λ−(〈F〉) is
continuous as well.
The theorem can fail if the limit sets are not disjoint. For example, consider the subsemi-
group of Aut(D) generated by two antiparallel loxodromic transformations g and h defined
as follows. Let σ be a reflection in the geodesic ` = [−i, i], and let σg and σh be reflections
in the geodesics `g = [z, 1] and `h = [z, 1] respectively, where z = eiθ for some θ ∈ (0, π/2).
3. ON SEMIGROUPS WITH DISJOINT LIMIT SETS 117
Let g = σgσ and h = σσh so that gh = σgσh is a parabolic map that fixes 1. Hence the
limit sets of the semigroup S = 〈g, h〉 meet at 1. However in every neighbourhood of g
there exists a map g1 such that g1h is elliptic of infinite order. Since g1 and h do not
commute, S1 = 〈g1, h〉 is dense in Aut(D) by [2], and so both limit sets of S1 (and in
particular Λ+(S1)) equal the whole ideal boundary. However Λ+(S) is a proper subset of
the ideal boundary. To see this, let Jg be the closed arc contained in the unit circle whose
end points are the landing points of `g and which contains αg. Similarly let Jh be the arc
contained in the unit circle whose end points are the landing points of σ(`h) and which
contains αh. It can be seen that both g and h map the set Jg ∪ Jh (the two highlighted
arcs in Figure 4.5) within itself, and so Λ+(S) ⊆ Jg ∪ Jh.
g
h
αg
αh
Jg
Jh
`
`h
`g
Figure 4.5. The generators g, h and the arcs Jg and Jh
Statements analogous to Theorem 4.23 can be found within the theory of complex dy-
namics. Let Rat(d) be the space of rational maps of degree d, endowed with the compact
open topology. A rational map f ∈ Rat(d) is hyperbolic if there exists a smooth conformal
metric defined on an open set containing the Julia set J(f), upon which f is expanding.
It can be shown that for a hyperbolic map f , there is some neighbourhood of f in Rat(d)
(endowed with the compact-open topology) upon which J(f) moves continuously with f .
See Branner and Fagella [9, Section 3.4] for further details. For semigroups, (see [15, sec-
tion 5]) the authors show that if S acts on C, has disjoint limit sets, and is generated by
finitely many non-elliptic maps, then S enjoys a property analogous to hyperbolicity for
118 4. LIMIT SETS
rational maps. Specifically, there is an open set containing Λ− and a metric defined on
this set, upon which each element in S is expanding. The metric they define is indeed
smooth and conformal. Theorem 4.23 shows that the limit sets of such semigroups vary
continuously as the generating set is perturbed, at least within some small neighbourhood.
Theorem 4.23 begs the question: What is the domain of stability of Λ+ and Λ−? That is for
which F0 ∈ Aut(B3)k is the map F 7→ Λ+(〈F〉) continuous in some neighbourhood of F0?
Indeed, one can ask similar questions with respect to other properties on semigroups. For
example, is the property of being semidiscrete and inverse free stable under perturbation
of the generators; in other words is the setF ∈ Aut(B3)k | S is semidiscrete and inverse free
open in Aut(B3)k ?
We can however deduce that the setF ∈ Aut(B3)k | S is semidiscrete and inverse free and Λ+(S) ∩ Λ−(S) = ∅
is open, for each positive integer k. To see this suppose F ∈ Aut(B3)k is such that 〈F〉
has disjoint limit sets and is semidiscrete and inverse free. Then 〈F〉 contains no ellip-
tic elements or the identity, and so by Theorem 4.23 we can perturb the generators in
some neighbourhood of F and preserve the property that the semigroup they generate
has disjoint limit sets. If the perturbation is sufficiently small, then the generators will
not contain elliptic maps or the identity, and so by Theorem 4.16 they must generate a
semidiscrete and inverse free semigroup.
We can still say something about how the limit sets vary with F , even if the limit sets of
S are not disjoint, as the next result shows. Throughout the proof of the next theorem
it is convenient to adjust our notation slightly, and denote the attracting fixed point of a
loxodromic Mobius transformation f by α(f) instead of αf . For a set X ⊆ S2 and ε > 0
we let (X)ε denote the set of points in S2 whose chordal distance from any point in X is
less than ε.
3. ON SEMIGROUPS WITH DISJOINT LIMIT SETS 119
Theorem 4.24. Suppose k ∈ N and let F ∈ Aut(B3)k be a bounded generating sequence
generating a nonelementary semigroup. Then for all ε > 0 there exists an open neighbour-
hood U of F such that for all G ∈ U ,
Λ+(〈F〉) ⊆ (Λ+(〈G〉))ε.
Proof. For each G ∈ Aut(B3)k let πG be the projection map from k<ω to the associ-
ated Mobius transformation in 〈G〉. That is, if G = (g1, . . . , gk) then
πG : k<ω → Aut(B3)
is given by
π((i0, . . . , iN−1)) = gi0 · · · giN−1
for each (i0, . . . , iN−1) ∈ k<ω. Since 〈F〉 is nonelementary its set of attracting fixed points
is dense in its forward limit set, and so we can choose a finite set of attracting fixed points
such that the open ε/2-balls centred at these points cover Λ+. For each of these attracting
fixed points α, choose a word t ∈ k<ω such that πF (t) has attracting fixed point α. We
let Tε denote the set of all such t, and so we have
Λ+(〈F〉) ⊆⋃t∈Tε
B(α(πF (t)), ε/2).
Since Tε is finite, there is an open neighbourhood of F , U say, such that for all G ∈ U and all
t ∈ Tε the Mobius transformation πG(t) is loxodromic, and the point α(πG(t)) is no further
than ε/2 from α(πF (t)). This is possible because the collection of loxodromic transforma-
tions is open in Aut(B3), and the attracting fixed point of a loxodromic Mobius transfor-
mation f varies continuously as f does. It follows that B(α(πF (t)), ε/2) ⊆ B(α(πG(t)), ε)
for all t ∈ Tε. Now since B(α(πG(t)), ε) ⊆ (Λ+(〈G〉))ε it follows that
Λ+(〈F〉) ⊆ (Λ+(〈G〉))ε
as required.
The conclusion of the theorem can be regarded as ‘one half’ of continuity. For suppose
the following statement were true: for all ε > 0 there exists some neighbourhood U of F
120 4. LIMIT SETS
such that for all G ∈ U ,
(16) Λ+(〈G〉) ⊆ (Λ+(〈F〉))ε.
When combined with the conclusion of the theorem, we could then infer that the map
from Aut(B3)k to the space of compact subsets of S2 given by
F 7→ Λ+(〈F〉)
is continuous. In fact condition (16) is false in general, as the example following Theo-
rem 4.23 demonstrates.
3.3. Uniqueness of generating sets. Let S = 〈F〉 where F is a finite set. This
subsection addresses the question of when F is uniquely determined by S. We say F is
a minimal generating set if it has no redundant elements, that is, there are no elements
we can remove from F and still generate S from the remaining elements. We first give
sufficient conditions that ensure S has a unique minimal generating set.
Theorem 4.25. Suppose S is generated by the finite minimal generating set F , and has
empty group part. Then F is the unique minimal generating set if and only if the semigroup
fSf−1 does not meet S−1 for all f ∈ F .
Proof. First suppose S has two different minimal generating sets for S, say F =
f1, . . . , fm and G = g1, . . . , gn. Then some element of F , which we take to be f1, does
not lie in G. Hence we can write f1 as a word gi1 . . . gik in G with length k ≥ 2. Now since
each gij lies in S, we can substitute gij for some word in F . We now have written f1 as a
word in F , which we denote by w, and has length at least two. It is important to empha-
sise that no cancellations have been carried out; we regard words merely as sequences of
letters, rather than sequences of Mobius transformations. If f1 is the first or last letter of
w, then f1 can be cancelled from both sides leaving a nonempty word in F equal to the
identity, which contradicts that I /∈ S. Yet if the letter f1 does not feature as a letter in
w, then f1, . . . , fm is not a minimal generating set, contrary to assumption. Hence we
must have f1 = w1f1w2 where w1, w2 are nonempty words in F . Hence w−11 = f1w2f−11
so that f1Sf−11 meets S−1.
3. ON SEMIGROUPS WITH DISJOINT LIMIT SETS 121
For the other direction, suppose fSf−1 meets S−1 for some f ∈ F , say f1w2f−11 = w−11
where w1 and w2 also lie in S. Hence f1 = w1f1w2. Then w1f1, w2, f2, . . . , fm is a
generating set that does not contain f1. Hence any minimal generating set contained in
w1f1, w2, f2, . . . , fm is a minimal generating set that does not contain f1.
The proof of the next corollary sheds light on the property fSf−1∩S−1 = ∅ for all f ∈ F ,
and the corollary itself serves as further evidence that supposing a semigroup’s limit sets
are disjoint is a strong property on semigroups.
Corollary 4.26. Suppose S is generated by a finite minimal generating set F that con-
tains no elliptic transformations. If the limit sets of S are disjoint then F is a unique
minimal generating set.
Proof. That S is semidiscrete and inverse free follows from Theorem 4.16, and so no
element in S is elliptic. If F were not a unique generating set, then fSf−1 would meet
S−1, say fgf−1 = h−1. If g is loxodromic then so is h, and moreover f maps the axis of
g to the axis of h, sending αg to βh and βg to αh. In particular the limit sets of S meet
contrary to assumption. If g is parabolic then again, the limit sets of S meet, contrary to
assumption. As g cannot be elliptic, F must be a unique generating set after all.
We do not know if this implication runs in the other direction; however we recall the
example given in Section 3 of Chapter 2 which is consistent with the converse.
In [15, Section 5] the authors show that whenever S is generated by a finite set F acting
on B3 such that each generator is not elliptic, then F may be regarded as a contracting
iterated function system acting on Λ+. They do this by considering the finitely many
connected components of Ω− that meet Λ+, and, as the limits sets are disjoint, these
cover Λ+. They then construct a metric derived from the hyperbolic metrics on each of
these components. In general S is not contracting on Λ+ with respect to the chordal
metric; however we can use Lemma 4.22 and ideas from the proof of Theorem 4.23 to find
a finite set FM ⊆ S such that SM = 〈FM 〉 has the same forward limit set as S, and FMis a contracting iterated function system with respect to the chordal metric, with limit
set Λ+. To do this, we recall that each loxodromic f ∈ Aut(B3) maps the interior of
122 4. LIMIT SETS
S2 \ I+(f) onto I−(f). Crucially each f is contracting with respect to the chordal metric
in the interior of S2 \ I+(f). By Lemma 4.22 each f ∈ FM has small isometric discs (of
radius less than 1/ sinh[12M ]). Since βf ∈ I+(f), the disc I+(f) is close to Λ−; similarly
αf ∈ I−(f). Hence for M large enough, no I−(f) meets I−(g) for any f, g ∈ FM . It
follows that FM is contracting on
S2 \⋃
f∈FM
I+(f)
and has forward limit set Λ+(S).
Collecting results from this chapter, we list some further properties of the class of semi-
group considered above.
Theorem 4.27. Let S be a semigroup generated by a finite set of Mobius transformations
that are not elliptic, and such that the limit sets of S are disjoint. Then
(i) S is discrete,
(ii) S has empty group part,
(iii) both Λ+(S) and Λ−(S) vary continuously in a neighbourhood of F ,
(iv) S has a unique minimal generating set, and
(v) every point in Λ+(S) is the limit point of some composition sequence, which converges
conically.
Proof. By Theorem 4.16, S is semidiscrete and inverse free, hence by Theorem 4.21
S is discrete. Now (iii) and (iv) follow by Theorem 4.23 and Corollary 4.26 respectively.
Finally (v) follows by Theorem 4.18.
CHAPTER 5
Self-maps of the disc
1. Introduction
This chapter is about composition sequences generated by finite subsets of the class
Mob1(D) of those Mobius transformations that map the unit disc strictly within itself.
If F1, F2, . . . is such a sequence, then the discs
D ⊇ F1(D) ⊇ F2(D) ⊇ · · ·
are nested, and the intersection⋂Fn(D) is either a single point or a closed disc (by disc
we mean a disc of positive radius in C, using the chordal metric). In the first case we
say that the composition sequence is of limit-point type, and in the second case we say
that it is of limit-disc type. Composition sequences have been much studied within the
literature on continued fractions (see, for example, [1, 4, 5, 27, 28]) – and the distinction
between limit-point type and limit-disc type is of central importance. We will see that, for
such composition sequences, limit-disc type really is quite special, and we find necessary
and sufficient conditions for limit-disc type to occur. It is already known (see for example
[27]) that every composition sequence of the type considered here converges ideally. Here
we prove more, that every composition sequence has a strong convergence property. A
composition sequence is of limit-point type exactly when the composition sequence con-
verges uniformly on the unit disc. If the composition sequence is of limit-disc type, then
the composition sequence merely converges locally uniformly (and not uniformly) on the
unit disc.
To state our first result we introduce a new concept: a composition sequence Fn generated
by a subset of Mob1(D) is said to be of limit-tangent type if all but a finite number of discs
from the sequence D ⊇ F1(D) ⊇ F2(D) ⊇ · · · share a single common boundary point.
123
124 5. SELF-MAPS OF THE DISC
Theorem 5.1. Let F be a finite subset of Mob1(D). Any composition sequence generated
by F of limit-disc type is of limit-tangent type.
That a composition sequence Fn is of limit-tangent type implies that there is a point p
and a sequence z1, z2, . . . in ∂D such that p = Fn(zn) for sufficiently large values of n. It
follows that fn(zn) = zn−1 for large n. We know that fn(D) ( D, so we deduce that fn(D)
is internally tangent to D at the point zn−1. Let us now choose any element f of Mob1(D):
the disc f(D) is either internally tangent to D at a unique point, or else it does not touch
the boundary of D. In the former case, we define uf and vf to be the unique points in ∂D
such that f(uf ) = vf , and in the latter case, we define uf = 0 and vf = ∞ (for reasons
of convenience to emerge shortly). The preceding theorem tells us that in order for the
sequence Fn to be of limit-disc type, we need ufn−1 = vfn for sufficiently large values of
n. How likely one is to find such a composition sequence is best illustrated by means of
a directed graph T (F), which we call the tangency graph of F , and which is defined as
follows. The vertices of T (F) are the elements of F . There is a directed edge from vertex
f to vertex g if uf = vg. Clearly, any vertex f for which f(D) and D are not internally
tangent is an isolated vertex.
Let us consider an example. One can easily check that a Mobius transformation f(z) =
a/(b + z), where a 6= 0, has the property that f(D) is contained within and internally
tangent to D if and only if |b| = 1 + |a|. Let
g(z) =12
32 + z
, h(z) =12
−32 + z
, and k(z) =−1
2
−32 + z
.
Each of these maps satisfies the condition |b| = 1+ |a|. Observe that g(−1) = 1, h(1) = −1
and k(1) = 1, which implies that ug = −1, vg = 1, uh = 1, and so forth. The tangency
graph of g, h, k is shown in Figure 5.1.
From the tangency graph we can see that a composition sequence Fn = f1 · · · fn gen-
erated by g, h, k is not of limit-tangent type if and only if (fn, fn+1) is equal to ei-
ther (g, g), (h, h), (g, k), or (k, h) for infinitely many positive integers n.
1. INTRODUCTION 125
kg
h
Figure 5.1. The tangency graph of g, h, k
We have shown that in order for a composition sequence Fn to be of limit-disc type,
the sequence of vertices f1, f2, . . . in the tangency graph must eventually form an infinite
path. The converse fails though: not all infinite paths in the tangency graph correspond to
sequences of limit-disc type. To determine which paths arise from composition sequences
of limit-disc type, we need to look at the derivatives of the maps fn at tangency points.
Let γf = 1/|f ′(uf )|, where f ∈ Mob1(D). The next theorem gives necessary and sufficient
conditions for a composition sequence to be of limit-disc type.
Theorem 5.2. Let F be a finite subcollection of Mob1(D). A composition sequence Fn =
f1 · · · fn generated by F is of limit-disc type if and only if:
(i) ufn−1 = vfn for all but finitely many positive integers n; and
(ii)
∞∑n=1
γf1 · · · γfn < +∞.
For the example illustrated in Figure 5.1, one can check that γg = γh = γk = 12 . It
follows that any composition sequence generated by g, h, k satisfies condition (ii) of
Theorem 5.2, so such a sequence is of limit-disc type if and only if it is of limit-tangent
type.
Next we turn to the following question: given a finite subcollection F of Mob1(D), how
many of the composition sequences generated by F are of limit-disc type? Using the
tangency graph we can formalise this question, and answer it, at least in part, using
known techniques. More precisely, we will define a metric on the set Fω of all sequences
126 5. SELF-MAPS OF THE DISC
taking values in F . Each x ∈ Fω gives rise to a composition sequence, the sequence with
nth term x(1) · · ·x(n).
Suppose (X, d) is any metric space and Y is a subset of X. We let |Y | denote the diameter
of Y . If Vi is some finite or countable family of sets where Vi ⊆ X and⋃i Vi contains
Y , then we say the family Vi is a cover of Y . If moreover |Vi| ≤ δ for each i, then we
say Vi is a δ-cover of Y . Recall that (see for example [13, Chapter 2]) if s ≥ 0 then we
define the quantity
Hsδ(Y ) = inf
∑i
|Ui|s | Ui is a δ-cover of Y
,
and define the Hausdorff s-measure of Y as
infδ→0Hsδ(Y ).
The Hausdorff dimension of Y is defined by
dim(Y ) = inf s ≥ 0 | Hs(Y ) = 0 = sup s ≥ 0 | Hs(Y ) = +∞ .
We evaluate the Hausdorff dimension of Λ(F), defined to be the set of elements in Fω
whose associated composition sequences are of limit-disc type. The value of the Hausdorff
dimension depends on the metric we choose; a standard choice for a metric on the set of
sequences from a finite alphabet (which we adopt) is as follows. Given sequences x and y
in Fω, we define
d(x, y) =1
|F|m, where m = mini ∈ N | x(i) 6= y(i).
Then (Fω, d) is a compact metric space.
A cycle in a directed graph is a sequence x0, . . . , xn of vertices in the graph such that
x0 = xn and there is a directed edge from xi−1 to xi, for i = 1, . . . , n. Recall that the
spectral radius of a square matrix is the largest modulus of the eigenvalues of the matrix.
Let ρ(F) denote the spectral radius of the adjacency matrix of T (F). We can now state
our first theorem on Hausdorff dimension.
1. INTRODUCTION 127
Theorem 5.3. Let F be a finite subcollection of Mob1(D), and suppose that γf < 1 for
f ∈ F . If F contains a cycle, then
dim Λ(F) =log ρ(F)
log |F|,
and otherwise dim Λ(F) = 0.
Returning again to the example illustrated in Figure 5.1, one can check that the spectral
radius of the adjacency matrix of the tangency graph of G = g, h, k is 12(1 +
√5). Also,
γg, γh, γk < 1. Therefore dim Λ(G) = (log(1 +√
5)− log 2)/ log 3.
As we will see, Theorem 5.3 is a corollary of Theorem 5.2 and a well-known result on
Hausdorff dimensions of paths in directed graphs. Also, because it assumes that condition
(ii) from Theorem 5.2 is satisfied, it is really about the Hausdorff dimension of the set
of composition sequences of limit-tangent type. In contrast, the next theorem determines
dim Λ(F) under the assumption that condition (i) from Theorem 5.2 is satisfied.
Theorem 5.4. Let F be a finite subcollection of Mob1(D), and suppose that T (F) is
the complete graph on |F| vertices. Then dim Λ(F) = 0 if γf ≥ 1 for all f ∈ F ;
dim Λ(F) = 1 if γf < 1 for all f ∈ F ; and otherwise
dim Λ(F) =log(
min∑
f∈F γ−sf | s ≥ 0
)log |F|
.
The problem of finding dim Λ(F) for an arbitrary finite subset F of Mob1(D) remains
open.
So far we have focused on determining those composition sequences of limit-disc type, and
we have ignored questions of convergence of sequences in the unit disc. In fact, it is known
already that any composition sequence generated by a finite subset of Mob1(D) converges
locally uniformly in D to a constant (this follows, for example, from [27, Theorems 3.8
and 3.10]). Here we will prove some stronger theorems on convergence of composition
sequences generated by a finite set.
Recall that a sequence of Mobius transformations F1, F2, . . . is called an escaping sequence
if ρ(ζ, Fn(ζ))→∞ as n→∞ for any choice of the point ζ in H3. We say that an escaping
sequence F1, F2, . . . is of convergence type if∑
exp(−ρ(j, Fn(j))) < +∞. This terminology
128 5. SELF-MAPS OF THE DISC
is borrowed from the theory of Kleinian groups, and indeed, a sum of the type above will
be familiar to Kleinian group theorists: it relates to the critical exponent of a Kleinian
group, a connection that has been explored before in [35].
Our first theorem on convergence demonstrates that composition sequences generated by
finite subsets of Mob1(D) are of convergence type. In fact, it is more general than this,
as it does not assume that the generating set is finite. Let rad(D) denote the Euclidean
radius of a Euclidean disc D.
Theorem 5.5. Let f1, f2, . . . be a sequence of Mobius transformations such that fn(D) ⊆ D
and rad(fn(D)) < δ, for n = 1, 2, . . . , where δ satisfies 0 < δ < 1. Then Fn = f1 · · · fn is
of convergence type.
The convergence type condition is a strong convergence property. By placing further,
relatively mild conditions on sequences of convergence type (such as assuming that the
sequence is generated by a finite set) one can prove that the sequence converges ideally.
What is more, if Fn is of convergence type and converges ideally to a point p, then one
can show that the set of points z in C such that Fn(z) does not converge to p as n→∞
has Hausdorff dimension at most 1 (see [35, Corollary 3.5]). For composition sequences
generated by a finite set and of the type that interest us, we have the following result.
Theorem 5.6. Let Fn = f1 · · · fn be a composition sequence generated by a finite subcol-
lection F of Mob1(D). Then Fn converges ideally to a point p in D. Furthermore, Fn
converges locally uniformly to p on D \X, where
X = uf ∈ ∂D | fn = f for infinitely many integers n.
Theorem 5.6 is just a corollary of Theorem 5.5, and the part about converging ideally is
known from other more general results (again, we refer the reader to [27, Theorems 3.8
and 3.10]). What we wish to emphasise is, first, the complete understanding of the dy-
namics that we have obtained, and, second, the interaction between the action of the
transformations in hyperbolic space and on its boundary, which is illuminated in the
proofs of the above pair of theorems. Although the results in this chapter are framed
in terms of the unit disc, by conjugation they hold for any disc on the Riemann sphere.
Indeed our results readily generalise to higher dimensions.
2. EVENTUALLY TANGENT SEQUENCES OF DISCS 129
2. Eventually tangent sequences of discs
In this section we prove Theorem 5.1. We make the elementary observation that if distinct
discs D and E satisfy D ⊆ E, then they can have at most one point of tangency.
Lemma 5.7. Let D1 ⊆ D2 ⊆ · · · be a nested sequence of discs in the plane, no two of
which are equal. The sequence is of limit-tangent type if and only if Dn is tangent to Dn+2
for all but finitely many values of n.
Proof. Suppose that the sequence D1, D2, . . . is eventually tangent. Then there is a
point p in the plane and a positive integer m such that p ∈ ∂Dn when n ≥ m. Therefore
Dn and Dn+2 share a common boundary point when n ≥ m, and because the two discs
are distinct and satisfy Dn+2 ⊆ Dn, they must be tangent to one another.
Conversely, suppose that there is a positive integer m such that Dn is tangent to Dn+2
when n ≥ m. For any particular integer k, with k ≥ m, let p be the point of tangency of
Dk and Dk+2, and let c and r be the Euclidean centre and radius of Dk+1. As the discs
D1, D2, . . . are nested, we see that first, p ∈ ∂Dk, so |p− c| ≥ r, and second, p ∈ ∂Dk+2,
so |p − c| ≤ r. Therefore |p − c| = r, which implies that Dk, Dk+1 and Dk+2 have a
(unique) common point of tangency. It follows that all the discs Dn have a common point
of tangency when n ≥ m.
We remark that the lemma remains true if we replace Dn and Dn+2 with Dn and Dn+q,
for any positive integer q > 1, but is false for q = 1.
Proof of Theorem 5.1. We prove the contrapositive; that is, we prove that if
F1(D) ⊆ F2(D) ⊆ · · · is not eventually tangent, then F1, F2, . . . is of limit-point type.
Suppose then that F1(D) ⊆ F2(D) ⊆ · · · is not eventually tangent. By Lemma 5.7, there is
an infinite sequence of positive integers n1 < n2 < · · · , where ni+1 > ni+1 for each i, such
that Fni+2(D) is not tangent to Fni(D) for each integer ni. Equivalently, fni+1fni+2(D)
is not tangent to D for each integer ni. Let K be the union of all the sets fg(D), where
f, g ∈ F , such that fg(D) is not tangent to D. Since F is finite, K is a compact subset of
D.
130 5. SELF-MAPS OF THE DISC
Let us define g1 = f1 · · · fn1 and gi = fni+1 · · · fni+1 for i = 2, 3, . . . . Then, for i > 1,
gi(D) = fni+1 · · · fni+1(D) ⊆ fni+1fni+2(D) ⊆ K.
Let Gn = g1 · · · gn. Since each gi maps D within a compact subset of D, [4, Theorem 4.6]
tells us that Gn converges uniformly on D to a constant. That is, Gn is of limit-point type.
Since the sequence G1(D), G2(D), . . . is a subsequence of F1(D), F2(D), . . . , we deduce that
Fn is also of limit-point type.
3. Necessary and sufficient conditions for limit-disc type
We define the chordal derivative of a Mobius transformation f by
f#(z) = limw→z
χ(f(z), f(w))
χ(z, w).
The chordal derivative is related to the Euclidean derivative (see [7, Equation 2.1]) by the
equation
f#(z) =(1 + |z|2)|f ′(z)|
1 + |f(z)|2
for z 6= f−1(∞),∞.
In the ball model, the orientation-preserving isometries of B3 equipped with the Euclidean
metric are exactly those Mobius transformations that fix 0. In the upper half-space model,
the orientation preserving isometries of (C, χ) are exactly those Mobius transformations
that fix j.
Let us define
γf =1
f#(uf ),
where f ∈ Mob1(D). In fact, f#(uf ) happens to coincide with |f ′(uf )| (the Euclidean
derivative), but we use the chordal derivative because it is defined everywhere on C, and
could be used in a version of the next lemma in which D is replaced by a half-plane (say).
In this section we prove Theorem 5.2. Let K denote the open right half-plane of C and
suppose that h is a Mobius transformation that satisfies h(K) ⊆ K and h(∞) =∞. Then
h(z) = az + b, where a > 0 and Re[b] ≥ 0. Moreover, h(K) = K if and only if Re[b] = 0.
3. NECESSARY AND SUFFICIENT CONDITIONS FOR LIMIT-DISC TYPE 131
Lemma 5.8. Suppose that hn and Hn are sequences of Mobius transformations defined by
hn(z) = anz + bn, where an > 0 and Re[bn] > 0, and Hn = h1 · · ·hn. The sequence of
nested discs H1(K), H2(K), . . . is of limit-disc type if and only if
∞∑n=1
a1 · · · an−1Re[bn] < +∞.
We assume that the expression a1 · · · an−1 takes the value 1 when n = 1.
Proof. Observe that Hn(z) = a1 · · · anz +∑n
k=1 a1 · · · ak−1bk. Therefore
Hn(K) = tn + K, where tn =
n∑k=1
a1 · · · ak−1Re[bk];
the result follows immediately.
Whenever we have a composition sequence Fn = f1 · · · fn, it will be convenient to define
F0 equal to I. We define Π to be the (hyperbolic) convex hull of D in the upper half-space
model, that is the smallest (hyperbolically) convex set in H3 containing each geodesic that
lands at points in D.
Lemma 5.9. Suppose that fn is a sequence of Mobius transformations from Mob1(D), and
z0, z1, . . . is sequence of points from ∂D such that fn(zn) = zn−1 for each positive integer
n. Let Fn = f1 · · · fn. Then the nested sequence of discs F1(D), F2(D), . . . is of limit-disc
type if and only if∞∑n=1
sinh ρ(j, fn(Π))
F#n−1(zn−1)
< +∞.
Proof. The discs D and K both have chordal radius 1, so there is a chordal isometry
φn that maps D to K and maps zn to ∞, for n = 0, 1, . . . . Let hn = φn−1fnφ−1n and
Hn = h1 · · ·hn. Then Hn = φ0Fnφ−1n . Hence Hn(K) = φ0Fn(D), so F1(D), F2(D), . . .
is of limit-disc type if and only if H1(K), H2(K), . . . is of limit-disc type. Observe that
hn(∞) =∞ and hn ∈ Mob1(K). Therefore we can write hn(z) = anz + bn, where an > 0
and Re[bn] > 0. By applying the chain rule, and remembering that the maps φn are
chordal isometries, we see that
F#n (zn) = H#
n (∞) =1
a1 · · · an.
132 5. SELF-MAPS OF THE DISC
Next, since hn(K) = bn + K, a simple calculation in hyperbolic geometry shows that
sinh ρ(j, hn(Σ)) = Re[bn],
where Σ is the hyperbolic plane with ideal boundary ∂K. Observe that
ρ(j, fn(Π)) = ρ(φn−1(j), φn−1fnφ−1n (Σ)) = ρ(j, hn(Σ)),
so sinh ρ(j, fn(Π)) = Re[bn]. To conclude, we deduce from Lemma 5.8 that F1(D), F2(D), . . .
is of limit-disc type if and only if
∞∑n=1
sinh ρ(j, fn(Π))
F#n−1(zn−1)
< +∞,
as required.
Proof of Theorem 5.2. Statement (i) of Theorem 5.2 is equivalent to the assertion
that the sequence of discs F1(D) ⊆ F2(D) ⊆ · · · is eventually tangent. Therefore Theo-
rem 5.1 tells us that if F1, F2, . . . is of limit-disc type, then statement (i) holds. Thus,
we have only to show that, on the assumption that statement (i) holds, F1, F2, . . . is of
limit-disc type if and only if statement (ii) holds.
Let zn = ufn , for n = 1, 2, . . . , and z0 = vf1 . Then
fn(zn) = fn(ufn) = vfn = ufn−1 = zn−1,
for sufficently large values of n. In particular, zk = fk+1 · · · fn(zn), for k = 1, . . . , n − 1.
Also, by the chain rule,
γ1 · · · γn =1
f#1 (z1) · · · f#n (zn)=
1
F#n (zn)
.
Since F is finite, there is a positive constant k such that
1
k< sinh ρ(j, f(Π)) < k,
for any map f in F . It follows that
∞∑n=1
γf1 · · · γfn < +∞ if and only if∞∑n=1
sinh ρ(j, fn(Π))
F#n−1(zn−1)
< +∞.
It follows then, from Lemma 5.9, that F1, F2, . . . is of limit-disc type if and only if state-
ment (ii) holds, and this completes the proof.
5. COMPLETE TANGENCY GRAPH 133
4. Hausdorff dimension of the set of composition sequences of limit-disc type
In this section we prove Theorem 5.3. First we need some terminology, and a result
[16, Corollary 2.9], which we state here, in our notation.
Theorem 5.10. Let G be a directed graph with vertices 0, . . . , b− 1 that contains a cycle.
Then the Hausdorff dimension of the elements in 0, . . . , b− 1ω that are paths in G is
given bylog ρ(G)
log b,
where ρ(G) is the spectral radius of G.
We say x ∈ Fω is a path in T (F) if for each n ∈ N there is a directed edge in T (F) from
x(n) to x(n + 1). We say x ∈ Fω is eventually a path in T (F) if for some m ∈ N the
sequence with nth term x(n+m) is a path in T (F). We can now deduce Theorem 5.3.
Proof of Theorem 5.3. By Theorem 5.2, a composition sequence is of limit-disc
type if and only if the associated element in Fω is eventually a path in the tangency
graph. That is, Λ(F) is exactly the set of those elements in Fω that are eventually a
path in T (F). If T (F) contains no cycles then clearly Λ(F) is empty, and so has Hausdorff
dimension 0. Otherwise we appeal to Theorem 5.10 and infer that the set of elements in
Fω that are paths in T (F) has Hausdorff dimension log ρ(F)/ log b. Hausdorff dimension
enjoys a countable stability property, that is for any sequence of sets Xn ⊆ Fω, we have
dim⋃∞n=1Xn = supn∈N dimXn. As F is finite, by this countable stability property it
follows that the collection of elements in Fω that are eventually paths in T (F) also has
Hausdorff dimension log ρ(F)/ log b.
5. Complete tangency graph
In this section we turn to the proof of Theorem 5.4. Recall the metric d on Fω defined
by d(x, y) = 1/|F|m where m = mini ∈ N | x(i) 6= y(i). So far we have worked with
Hausdorff dimensions of subsets of the metric space (Fω, d). By associating composition
sequences generated by F with sequences in Fω, we were able to assign a Hausdorff di-
mension to the collection of composition sequences that are of limit-disc type. In this
section we represent composition sequences not by elements in Fω, but by points in [0, 1],
134 5. SELF-MAPS OF THE DISC
via their base |F|-expansion. We think of composition sequences as points in the metric
space [0, 1] endowed with the Euclidean metric. In doing so we give rise to a different
notion of Hausdorff dimension on sets of composition sequences. In fact, the two notions
of Hausdorff dimension coincide. A form of this statement can be deduced from [14, The-
orem 5.1], however for convenience we state and prove the result here, in our notation.
Let b = |F| and choose an enumeration f0, . . . , fb−1 of F . We now regard sequences
in F as sequences in 0, . . . , b− 1. Define I1 ⊆ [0, 1] as the set of points whose base b-
expansion is not eventually equal to 0, or eventually equal to b−1. Let B1 ⊆ 0, . . . , b− 1ω
denote those sequences that are not eventually equal to 0, or eventually equal to b − 1.
Then when the projection π : 0, . . . , b− 1ω −→ [0, 1] given by π(x) =∑∞
j=0 x(j)b−j is
restricted to B1, we obtain a bijection from B1 onto I1. Given any X ⊆ [0, 1] let Is(X)
be the Hausdorff s-measure of X with respect to the Euclidean metric on [0, 1]. Similarly,
for Y ⊆ 0, . . . , b− 1ω let Bs(Y ) be the Hausdorff s-measure of Y induced by the metric
space (0, . . . , b− 1ω , d). The quantities Isδ (X) and Bsδ(Y ) are defined in the obvious
way. Since both [0, 1] \ I1 and 0, . . . , b− 1ω \B1 are countable, Is(I1) = Bs(B1) = 1 for
all s > 0.
Lemma 5.11. For any X ⊆ [0, 1] and s > 0 we have that Is(X) = 0 if and only if
Bs(π−1(X)) = 0. Consequently the Hausdorff dimensions of X and π−1(X) are equal.
Proof. Suppose X ⊆ I1 and choose any s > 0. Let us write Y = π−1(X). If x, y ∈ B
are such that d(x, y) = b−m, then π(x) and π(y) lie in the same b-adic interval of level
m − 1. Hence |π(x) − π(y)| ≤ b−m+1 = bd(x, y), and so π is a Lipschitz map. Hence by
[14, Lemma 1.8] there exists c > 0 such that Is(π(Y )) ≤ csBs(Y ) for all Y ⊆ B1. In the
other direction, suppose Y ⊆ B1 and choose δ > 0 and ε > 0. Suppose Vi is a family
where each Vi ⊆ I1, the family forms a δ-cover of π(Y ), and satisfies∑i
|Vi|s ≤ Isδ (π(Y )) + ε.
For each i let ki be the unique positive integer such that
b−ki−1 ≤ |Vi| < b−ki .
5. COMPLETE TANGENCY GRAPH 135
Each Vi meets at most two b-adic intervals of level ki. If Vi meets two such intervals we
denote them by Vi0 and Vi1. Otherwise Vi meets one such interval, which we denote by
Vi0, and let Vi1 = ∅. Note that
|π−1(Vij)| ≤ b−ki−1
so that
(17) |π−1(Vij)| ≤ |Vi|.
Since Vij is a cover of π(Y ), the familyπ−1(Vij)
is a cover of Y , and by (17) it is a
δ-cover. Since |Vij | ≤ b−ki ≤ b|Vi| it follows that
Bsδ(Y ) ≤
∑i,j
|π−1(Vij)|s ≤∑i,j
|Vij |s ≤ 2bs∑i
|Vi|s
≤ 2bs(Isδ (π(Y )) + ε).
Letting ε→ 0 we obtain Bsδ(Y ) ≤ 2bsIsδ (π(Y )). Hence
Bsδ(Y ) ≤ 2bsIsδ (π(Y )) ≤ 2bsIs(π(Y )),
and letting δ → 0 yields Bs(Y ) ≤ 2bsIs(π(Y )).
Hence we have shown that
Bs(Y ) ≤ 2bsIs(π(Y )) ≤ 2bscsBs(Y )
for all Y ⊆ B1. Since B \B1 is countable, the above also holds for any Y ⊆ B. It follows
that for any X ⊆ [0, 1] we have Bs(π−1(X)) = 0 if and only if Is(X) = 0.
Above, we began by choosing an enumeration of F in order to convert sequences in F into
sequences in 0, 1, . . . , b− 1. Because of the nature of the metric d on 0, . . . , b− 1ω,
the Hausdorff dimension of subsets of 0, . . . , b− 1ω is invariant under permutations of
0, 1, . . . , b− 1. It follows that the Hausdorff dimension we defined on composition se-
quences via the interval [0, 1] is also independent of the choice of enumeration of F .
136 5. SELF-MAPS OF THE DISC
We now work towards a proof of Theorem 5.4. Suppose γ0, . . . , γb−1 ⊆ R+. For each
x ∈ 0, . . . , b− 1ω and i = 0, 1, . . . , b− 1 we write
Ni(x, n) = |j ∈ 1, . . . , n | x(j) = i|.
In light of the above lemma and Theorem 5.2, our goal in this section is to prove the
following.
Theorem 5.12. Suppose γ0, . . . , γb−1 ⊆ R+ is such that γi > 1 and γj < 1 for some
i, j. Let X be the π-image of those x ∈ 0, . . . , b− 1ω such that the series
(18)
∞∑n=1
n∏j=1
γx(j)
converges. Then
dimX = infs>0
log
(b−1∑i=0
γ−si
)/ log b.
It is easy to find dimX if γi ≤ 1 for all i, or if γi ≥ 1 for all i. For example in the
latter case, X is empty and dimX = 0. Hence to prove Theorem 5.4 it suffices to prove
Theorem 5.12. Given any x ∈ 0, . . . , b− 1ω we define
log ηn(x) =
b−1∑i=0
Ni(x, n)
nlog γi,
and relate the convergence of the series
∞∑n=1
n∏j=1
γx(j)
to the sequence ηn(x).
Lemma 5.13. If lim supn→∞ log ηn(x) < 0 then the series
∞∑n=1
n∏j=1
γx(j)
converges, while, if lim supn→∞ log ηn(x) > 0, then the series diverges.
5. COMPLETE TANGENCY GRAPH 137
Proof. This result is no more than an application of the nth root test for converging
series. It follows from the test that
(19)∞∑n=1
n∏j=1
γx(j)
converges if lim supn→∞
(∏nj=1 γx(j)
)1/n< 1, but diverges if lim supn→∞
(∏nj=1 γx(j)
)1/n>
1. Noting that the logarithm function is increasing, a straightforward computation shows
that
log
1
nlim supn→∞
n∏j=1
γx(j)
= lim supn→∞
log ηn(x)
and so the result follows.
For b = 1, 2, . . . and x ∈ [0, 1] we write In(x) as the nth level b-adic half open interval
containing x. We shall use the following result, which is also known as Billingsley’s lemma,
which can be found in [13, Proposition 2.3].
Lemma 5.14. Suppose A ⊆ [0, 1] is a Borel set and let µ be a Borel measure on [0, 1] such
that µ([0, 1]) <∞. If for all x ∈ A we have
lim infn→∞
logµ(In(x))
log |In(x)|≤ s,
then dimA ≤ s. If µ(A) > 0 and for all x ∈ A we have
t ≤ lim infn→∞
logµ(In(x))
log |In(x)|,
then t ≤ dimA.
We say p = (p0, . . . , pb−1) is a probability vector if 0 ≤ pi ≤ 1 for each i, and∑
i pi = 1.
For i, j ∈ 0, 1, . . . , b− 1 we define
δi,j =
1 if i = j
0 if i 6= j.
Using Lemma 5.14, the following result [8, Equation 14.4] can be shown.
138 5. SELF-MAPS OF THE DISC
Lemma 5.15. Let p = (p0, . . . , pb−1) be a probability vector. Given x ∈ [0, 1] let x(j) denote
the j-th digit in the b-adic expansion of x. Let Fp be the set of x ∈ [0, 1] such that for each
i = 0, . . . , b− 1
limn→∞
1
n
n∑j=1
δx(j),i = pi.
Then
dimFp =−∑
i pi log pilog b
.
To prove Theorem 5.12 we first establish some properties of the function given by
g(s) = logb−1∑i=0
γ−si
on R. For convenience we define f(s) = −g′(s).
Lemma 5.16. Suppose that γ0, . . . γb−1 ⊆ R+ satisfy the assumptions of Theorem 5.12.
Then
lims→+∞
g(s) = lims→−∞
g(s) = +∞,
and g attains a unique global minimum in R. The function f is strictly decreasing and
has exactly one zero at the global minimum of g.
Proof. The hypotheses on γ0, . . . , γb−1 imply that g(s)→∞ as |s| → ∞ does, and
as g is continuous, g(s) | s ∈ R attains a minimum value g(s0) at some local minimum
s0 of g. Differentiating g gives
f(s) =
∑i γ−si log γi∑i γ−si
.
Note that as s → −∞ then for γ > 1, γ−s log γ → ∞ while if γ < 1 then γ−s log γ → 0.
Similarly as s → +∞ then for γ > 1, γ−s log γ → 0 while if γ < 1 then γ−s log γ → −∞.
As the denominator of f is always positive, it follows that f(s) takes negative values
near +∞ and positive values near −∞, and in particular f(s) has at least one zero. A
computation shows that
f ′(s) =1(∑i γ−si
)2(∑
i
(log γi)γ−si
)2
−
(∑i
(log γi)2γ−si
)(∑i
γ−si
)=f(s)2 −
∑i γ−si (log γi)
2∑i γ−si
,
5. COMPLETE TANGENCY GRAPH 139
so that f ′(s) < 0 if f(s) = 0; that is, f is decreasing at its zeros. In fact f is a strictly
decreasing function on R: an application of the Cauchy-Schwarz inequality yields(∑i
(log γi)γ−si
)2
=
(∑i
(γ−s/2i log γi)(γ
−s/2i )
)2
≤
(∑i
(log γi)2γ−si
)(∑i
γ−si
).
Moreover the inequality is strict, for otherwise (γ−s/20 , . . . , γ
−s/2b−1 ) is a multiple of
(γ−s/20 log γ0, . . . , γ
−s/2b−1 log γb−1), which implies that log γi takes the same value over i =
0, . . . , b − 1, contrary to our assumption on the γi. Since f is continuous it must have
exactly one zero, so that g(s) has exactly one local extremum in R, which is hence also a
global minimum.
Note that Lemma 5.16 says that the infimum of Theorem 5.12 is either attained as a
minimum for some s > 0, or is equal to g(0) = log b, in which case dim(X) = 1.
Let us denote by s0 the unique real number where g attains its minimum. We write
X<0 =
π(x) | x ∈ 0, . . . , b− 1ω \ E2 and lim sup
n→∞log ηn(x) < 0
,
and similarly define the set X≤0, so that by Lemma 5.13 we have the inclusion
(20) X<0 ⊆ X ⊆ X≤0.
We can now prove Theorem 5.12.
Proof of Theorem 5.12. For s ∈ R we define µs to be the probability measure
generated by the probability vector
1∑i γ−si
(γ−s0 , . . . , γ−sb−1
).
Now, for x ∈ 0, . . . , b− 1ω, let In(x) denote the n-level b-adic interval containing π(x).
It follows that
logµs(In(x))
log |In(x)|=
log((∑
i γ−si
)−n∏i γ−sNi(x,n)i
)log b−n
=−n log(
∑i γ−si )− s
∑iNi(x, n) log γi
−n log b;
hence lim infn→∞
logµs(In(x))
log |In(x)|=
log(∑
i γ−si )
log b+ lim inf
n→∞
s
log blog ηn(x).(21)
140 5. SELF-MAPS OF THE DISC
We now show that dim(X) ≥ infs>0 log(∑
i γ−si
)/ log b. Let Fs denote the set of x ∈ [0, 1]
such that for each j = 0, . . . , b− 1
limn→∞
Nj(x, n)
n=
γ−sj∑i γ−si
.
If x ∈ Fs then
limn→∞
log ηn(x) =
∑i γ−si log γi∑i γ−si
= f(s).
By Lemma 5.16 we have f(s0 + ε) < 0 and so Fs0+ε ⊆ X by Lemma 5.13. Now applying
Lemma 5.15 gives
dim(Fs) =g(s)
log b+sf(s)
log b;
hence for all ε > 0 we have
dim(X) ≥ dim(Fs0+ε) =g(s0 + ε) + (s0 + ε)f(s0 + ε)
log b.
If s0 < 0 then set ε = −s0 to obtain dim(X) ≥ g(0)/ log(b) = 1, so that dim(X) = 1.
Otherwise s0 ≥ 0, and since f(s0 + ε)→ 0 as ε→ 0, we obtain the bound
dim(X) ≥ g(s0)
log b.
Finally we give the upper bound on dim(X). We know that dim(X) = 1 if s0 < 0, so we
need only consider s0 ≥ 0. Now if s ≥ 0 and x ∈ X≤0 then equation (21) gives
lim infn→∞
logµs(In(x))
log |In(x)|=g(s)
log b+
s
log blim infn→∞
log ηn(x)
≤ g(s)
log b+
s
log blim supn→∞
log ηn(x)
≤ g(s)
log b.
Now by Lemma 5.14 and by the set inclusion (20) we have
dim(X) ≤ dim(X≤0) ≤g(s)
log b.
Since g attains its minimum at s0 ≥ 0 we obtain the bound
dim(X) ≤ infs>0
g(s0)
log b.
Amalgamating our findings, if s0 ≥ 0 then we have dim(X) = g(s0)/ log b, while if s0 ≤ 0,
then dim(X) = g(0)/ log b = 1.
6. CONVERGENCE OF COMPOSITION SEQUENCES 141
6. Convergence of composition sequences
So far in this chapter we have concerned ourselves with the dichotomy between limit-point
and limit-disc type. We now turn to the issue of convergence, and in particular we prove
Theorem 5.5. Recall that the height of a point z + tj ∈ H3 is denoted by ht[z + tj] = t.
Lemma 5.17. Let U and V be distinct Euclidean discs with centres u and v, and radii r
and s, in that order. Suppose that V ⊆ U and
(22)r
s< min
2, 1 +
1
8sinh ρ(z,Π(V ))
,
for some point z in Π(U). Let w be the point in Π(V ) with ht[w] = ht[z] that is closest in
Euclidean distance to z. Then
|z − w| < 2(r − s).
Proof. Using a standard formula for the hyperbolic metric (see, for example, [3,
Section 7.20]) we have
sinh ρ(z,Π(V )) =|z − v|2 − s2
2hs,
where h = ht[z]. As V ⊆ U , we see that |u− v|+ s ≤ r. Hence
|z − v|2 − s2 ≤ (|z − u|+ |u− v|)2 − s2 ≤ (2r − s)2 − s2 = 4r(r − s).
Therefore, using (22), we obtain
sinh ρ(z,Π(V )) ≤ 2r(r − s)hs
=2r
h
(rs− 1)<
r
4hsinh ρ(z,Π(V )).
It follows that h < r/4 < s/2.
Now, let Σ denote the Euclidean plane in H3 with height h. Let U0 and V0 denote the
Euclidean discs in Σ with centres u+ hj and v + hj, and radii√r2 − h2 and
√s2 − h2, in
that order. Observe that V0 ⊆ U0 and z ∈ U0 and w ∈ V0. Using elementary Euclidean
geometry, we can see that
|z − w| < 2√r2 − h2 − 2
√s2 − h2.
142 5. SELF-MAPS OF THE DISC
But h < s/2 and h < r/2, so
2√r2 − h2 − 2
√s2 − h2 =
2(r2 − s2)√r2 − h2 +
√s2 − h2
<2√3
(r − s).
Therefore |z − w| < 2(r − s), as required.
Proof of Theorem 5.5. Let η = min
2, 1 + 18 sinh δ
and let rn be the Euclidean
radius of Fn(D) (with r0 = 1). Now define
A =
n ∈ N
∣∣∣∣ rn−1rn< η
and B =
n ∈ N
∣∣∣∣ rn−1rn≥ η
.
Suppose that n ∈ A. Observe that ρ(Fn−1(j), Fn(Π)) = ρ(j, fn(Π)) > δ, so
rn−1rn
< η < min
2, 1 +
1
8sinh ρ(Fn−1(j), Fn(Π))
.
Let wn be the point in Fn(Π) with ht[wn] = ht[Fn−1(j)] that is closest in Euclidean distance
to Fn−1(j). Lemma 5.17 tells us that
|Fn−1(j)− wn| < 2(rn−1 − rn).
Now, using a basic estimate of hyperbolic distance, we find that
δht[Fn−1(j)] < ρ(Fn−1(j), Fn(Π))ht[Fn−1(j)] ≤ ρ(Fn−1(j), wn)ht[Fn−1(j)] ≤ |Fn−1(j)−wn|.
Hence δht[Fn−1(j)] < 2(rn−1 − rn), so∑n∈A
ht[Fn−1(j)] < +∞.
Next, let n1 < n2 < . . . be the elements of B. As rnk/rnk−1≤ 1/η, we see that rnk ≤ 1/ηk.
Therefore
ht[Fnk−1(j)] ≤ rnk−1 ≤ rnk−1≤ 1
ηk,
so ∑n∈B
ht[Fn−1(j)] < +∞.
Observe that− log ht[Fn−1(j)] ≤ ρ(j, Fn−1(j)), from which it follows that exp[−ρ(j, Fn−1(j))] ≤
ht[Fn−1(j)]. We have just seen that
∞∑n=1
ht[Fn−1(j)] < +∞,
and hence we conclude that F1, F2, . . . is of convergence type.
6. CONVERGENCE OF COMPOSITION SEQUENCES 143
Proof of Theorem 5.6. Let us begin by proving that Fn converges ideally. In this
part of the proof it is convenient to switch to the unit ball model of hyperbolic space. In
this model the distinguished point 0 replaces the point j in H3. Two standard formulas
for the hyperbolic metric in B3 are
ρ(0, z) = log1 + |z|1− |z|
and sinh 12ρ(z, w) =
|z − w|√(1− |z|2)(1− |w|2)
.
From the first standard formula we see that 1 − |z| = (1 + |z|)e−ρ(0,z). Substituting this
equation into the second standard formula, and noting that 1 + |z| ≤ 2, we obtain
|z − w| =√
(1− |z|2)(1− |w|2) sinh 12ρ(z, w) ≤ 2(e−ρ(0,z) + e−ρ(0,w)) sinh 1
2ρ(z, w).
We apply this formula with z = Fn−1(0) and w = Fn(0). Let k > 0 be such that
ρ(0, fn(0)) < k for every positive integer n (as there are only finitely many different maps
fn). Then ρ(Fn−1(0), Fn(0)) < k for each integer n. Therefore
|Fn−1(0)− Fn(0)| < 2 sinh 12k(exp(−ρ(0, Fn−1(0))) + exp(−ρ(0, Fn(0))).
Theorem 5.5 tells us that Fn is of convergence type, so we can see that∑|Fn−1(0) −
Fn(0)| < +∞. It follows that F1(0), F2(0), . . . converges in the Euclidean metric on B3.
This sequence cannot converge to a point in B3, because Fn is an escaping sequence.
Therefore Fn converges ideally.
Let us now go back to thinking about Fn acting on C and H3. We have seen that Fn
converges ideally to a point p. This point p must belong to D because the orbit Fn(j) is
constrained within the Euclidean interior of Π.
It remains to show that Fn converges locally uniformly to p on D \ X, where X is the
finite set X = uf ∈ ∂D | fn = f for infinitely many integers n. To prove this, we first
note that it follows from Theorem 1.1 that Fn converges locally uniformly to p on the
complement of its backward limit set. In our circumstances, the backward limit set is
contained in the complement of D. Therefore Fn converges locally uniformly to p on D.
Let us choose q ∈ D \ X. Then there is a compact subset K of D such that f(q) ∈ K
for all f ∈ F . By increasing the size of K slightly (still a compact subset of D) we can
assume that there is an open Euclidean disc Q centred on q that does not intersect X such
144 5. SELF-MAPS OF THE DISC
that f(Q) ⊆ K for all f ∈ F . It follows that Fn converges locally uniformly to p on the
complement of X in D, as required.
CHAPTER 6
Coding limit sets
This contents of this chapter are work in progress, and the author expects that some re-
sults are likely to hold in a more general setting than the one described here.
Consider the semigroup generated by the two parabolic transformations f0(z) = z+ 1 and
f1(z) =z
z + 1in Aut(H2). Then the forward limit set is equal to the interval [0,∞] ⊆ R.
Moreover for every irrational number x in [0,∞] there is a unique composition sequence
generated by f0 and f1 that converges ideally to x. Indeed, any composition sequence that
converges ideally to a point x is essentially the usual continued fraction expansion of x of
the form
x = a1 +1
a2 +1
a3 + · · ·
.
In this chapter we consider semigroups such that, for some points in the semigroup’s
forward limit set (and as we shall see, most points), there exists more than one composition
sequence that converges ideally to that point. Both for ease of analysis and exposition
we work with a narrow class of semigroup. We choose two loxodromic elements f0 and
f1 in Aut(D), such that the set of attracting fixed points αf0 , αf1 does not meet the
set of repelling fixed points βf0 , βf1, and both repelling fixed points lie in the same
component of S1 \ αf0 , αf1. Let σv denote a reflection in the imaginary axis and let
σh denote a reflection in the real axis. The axes of f0 and f1 either meet in D, or not.
In both cases we can conjugate the semigroup such that the axes of the generators enjoy
the following symmetries: both axes ax(f0) and ax(f1) are fixed setwise by σh, and σv
transposes ax(f0) and ax(f1). Examples of the two cases are shown in Figure 6.1. In both
145
146 6. CODING LIMIT SETS
cases the connected component of S1 \ αf0 , αf1 that does not contain either βf0 or βf1 ,
shown in red, is mapped strictly within itself by both generators. Hence S is semidiscrete
and inverse free by Theorem 2.5, and the forward limit set is contained in the red interval,
by Theorem 1.8 applied to S−1. Similarly it follows that, in both cases, the backward
limit set of S is contained in the blue interval.
f1f0
f1f0
Figure 6.1. Possible configurations for the axes of f0 and f1
Let S be the semigroup generated by f0 and f1. Let 0, 1ω denote the set of all sequences
taking values in 0, 1, and let 0, 1n denote the set of all functions from 0, 1, . . . , n− 1
to 0, 1. We write
0, 1<ω =∞⋃n=0
0, 1n
as the set of all finite sequences taking values in 0, 1. In a similar manner to the
proof of Lemma 4.22, we endow 0, 1<ω with the usual partial order relation; that is for
i, j ∈ 0, 1ω then i ≤ j if and only if the domain of i is contained in the domain of j and
i agrees with j on the domain of i. Hence we can regard 0, 1<ω as a tree.
For any i ∈ 0, 1<ω with domain 0, 1, . . . , n− 1 we write |i| = n. For any m ≤ |i| we let
i m = (i0, . . . , im−1) be the restriction of i to 0, 1, . . . ,m− 1. For any i ∈ 0, 1n we
abbreviate the composition sequence fi0 · · · fin−1 to Fi, and we define F∅ to be the iden-
tity. For i and j in 0, 1<ω we let i, j denote the concatenation of i and j. We call the
elements (i0, . . . , i|i|−1, 0) and (i0, . . . , i|i|−1, 1) in 0, 1<ω the children of i = (i0, . . . , i|i|−1).
Let X denote the closed interval in S1 bounded by the attracting fixed points of the gener-
ators that does not contain the generators’ repelling fixed points. Since S is semidiscrete
6. CODING LIMIT SETS 147
and inverse free it follows from Theorem 4.18 that for any i ∈ 0, 1ω the composition
sequence with nth term Fin = fi0 · · · fin−1 converges ideally to a point, which we shall
denote by x(i). It follows from Theorem 1.1 that the sequence Fin(X) converges to the
singleton x(i). Indeed, Fin(X) is a decreasing nest of sets since Fin is a composition
sequence. If for i ∈ 0, 1ω the sequence Fin converges ideally to x, we say i is an address
of x. Theorem 4.18 also tells us that each point in the forward limit set is the limit point
of some composition sequence generated by F . In other words, each point in Λ+ has an
address. Let Y denote the set f0(X) ∩ f1(X), and let us write a = αf0 and b = αf1 . If
Y is empty then each point in Λ+ has a unique address, as is the case in the familiar
Cantor semigroup (although, strictly speaking, the Cantor semigroup is not an example of
the type of semigroup we consider here as its generators’ repelling fixed points coincide).
Since X is forward invariant under S, the forward limit set is certainly contained within
X, and indeed Λ+ = X if and only if Y is not empty. As we shall refer to them frequently,
for any j ∈ 0, 1<ω we abbreviate Fj(X) as Xj and Fj(Y ) as Yj .
In this chapter we focus on the case where Y has a nonempty interior, and so each point
x ∈ X has some address, which is not necessarily unique. We shall classify points in S1
by how many addresses they have. Clearly any point not in X has no address, while the
points a and b always have exactly one address.
For a cardinal κ we define Eκ as the set of x ∈ S1 such that the set of addresses for x
has cardinality κ. We write E<κ as the set of z ∈ S1 such that i ∈ 0, 1ω | x(i) = z
has cardinality less than κ. The sets E>κ and E≤κ are similarly defined. Our first goal
is to show that the set of points in X with infinitely many addresses, that is E≥ℵ0 , is a
‘topologically large set’. Towards this we first give another description of E≥ℵ0 . Recall
that we say a branch of a tree is cofinal if it intersects every level of the tree.
Lemma 6.1. Let f0 and f1 be two loxodromic elements in Aut(D) that are not antiparallel,
and let X and Y be the sets described above, which depend on f0 and f1. If Y is non-
empty, then Λ+(〈f0, f1〉) = X. If Y has a non-empty interior, then the set of points in X
148 6. CODING LIMIT SETS
with infinitely many addresses is given by
E≥ℵ0 =x ∈ X |
i ∈ 0, 1<ω | F−1i (x) ∈ Y
is infinite
=
∞⋂n=0
∞⋃k=n
i∈0,1k
Fi(Y ).
Proof. If Y is non-empty then X = f0(X) ∪ f1(X), so that for each x ∈ X there
exists j ∈ 0, 1 such that x ∈ fj(X). Hence we can recursively choose i ∈ 0, 1ω such
that f−1in−1· · · f−1i1 f
−1i0
(x) ∈ X, that is x ∈ fi0fi1 · · · fin−1(X) for each n ∈ N. The com-
position sequence fi0fi1 · · · fin−1 = Fin converges ideally, and since X does not meet Λ−
this convergence is uniform on X. Since x ∈ Fin(X) for all n, it follows that Fin con-
verges ideally to x, and in particular, x lies in the forward limit set of 〈f0, f1〉. Hence
Λ+(〈f0, f1〉) = X as claimed.
Now suppose that Y has non-empty interior. Given each x ∈ X let T (x) be the set of i ∈
0, 1<ω such that F−1i (x) ∈ X. T (x) is a tree, since for i ∈ T (x), Xi ⊆ Xin for each n =
0, 1, . . . , |i|. Furthermore, as Y = f0(X)∩f1(X), if i belongs to T (x), then both its children
also belong to T (x) if and only if F−1i (x) ∈ Y . The tree T (x) has infinitely many vertices
since x has at least one address. If only finitely many vertices of T (x) have two children,
then as a subtree of 0, 1<ω, T (x) has only finitely many cofinal branches. Conversely if
infinitely many vertices in T (x) have two children, then T (x) has infinitely many cofinal
branches. It follows that E≥ℵ0 =x ∈ X |
i ∈ 0, 1<ω | F−1i (x) ∈ Y
is infinite
.
Finally we observe that∞⋂n=0
∞⋃k=n
i∈0,1k
Fi(Y )
is simply a set-theoretic description ofx ∈ X |
i ∈ 0, 1<ω | F−1i (x) ∈ Y
is infinite
.
In fact it can be shown that for any subtree of 0, 1ω, there are either finitely many,
countably infinitely many or continuum many cofinal branches. For x ∈ X, the tree T (x)
from the proof of Lemma 6.1 is a subtree of 0, 1ω, and so x either has finitely, countably
infinitely many or continuum many addresses. Hence Eκ is nonempty only for κ finite,
6. CODING LIMIT SETS 149
κ = ℵ0 and κ = 2ℵ0 .
Recall from Section 2.1 in Chapter 4 that a subset of a topological space is meagre if it is
the countable union of nowhere dense sets. A residual set is the complement of a meagre
set. Meagre sets are ‘topologically small’ sets – they form a σ-ideal, and so residual sets
can be regarded as ‘topologically large’ sets.
Corollary 6.2. Let f0 and f1 be two loxodromic elements in Aut(D) that are not an-
tiparallel, and let X be the set described above, that is the closed interval bounded by αf0
and αf1 that does not contain βf0 or βf1. Then E≥ℵ0 is a residual subset of X.
Proof. Since Y is a proper interval, E≥ℵ0 contains the Gδ set
U =∞⋂n=0
∞⋃k=n
i∈0,1k
Fi(Y),
where Y is the interior of Y . Since
∞⋃k=n
i∈0,1k
Fi(Y)
is open and dense in X, it follows from the Baire category theorem that U is dense in X,
and so E≥ℵ0 contains a dense Gδ subset of X. It follows that E≥ℵ0 is a residual set.
Recall the chordal metric defined in Equation 1 in Chapter 5. We next record two distor-
tion results, which can be found in [34, Equation 1.3.2] and [34, Theorem 1.3.4], that we
shall need again.
Lemma 6.3. In the disc model, for any Mobius transformation g fixing the unit disc and
any points z and w in S1, we have
(23)|g(z)− g(w)||z − w|
= |g#(z)|1/2|g#(w)|1/2
and
(24) |g#(z)| = 1− |g(0)|2
|z − g−1(0)|2.
150 6. CODING LIMIT SETS
The following notation will be useful in what follows. For two sequences of positive real
numbers, xn and yn, for n ∈ N, we write xn yn if there exists positive real constants A
and B such that A ≤ xn/yn ≤ B for all large enough n.
Lemma 6.4. We have |Xj | |Yj | over all j ∈ 0, 1<ω.
Proof. This statement depends on the fact that the backward and forward limit sets
of S = 〈f0, f1〉 are disjoint.
Since |Xj | = |Fj(a)− Fj(b)| we obtain
(25) |Xj | =(1− |Fj(0)|2)|X|
|F−1j (0)− a||F−1j (0)− b|
upon combining (23) and (24).
Similarly, we have
|Yj | =(1− |Fj(0)|2)|Y |
|F−1j (0)− f1(a)||F−1j (0)− f0(b)|and so
|Yj ||Xj |
=|Y ||F−1j (0)− a||F−1j (0)− b|
|X||F−1j (0)− f1(a)||F−1j (0)− f0(b)|.
Since the points a, b, f1(a) and f0(b) do not lie in the backward limit set of S, each term in
the numerator and denominator of the right-hand side is bounded away from 0, uniformly
over all j ∈ 0, 1<ω.
Corollary 6.2 shows that the set of x ∈ X that have infinitely many addresses is topologi-
cally a large set. The next theorem shows that this set is large in a measure theoretic sense
too. For any x ∈ C and r > 0 we shall use the notation Bx(r) to denote the Euclidean
open ball centred at x and of radius r. We let µ denote the Lebesgue measure on S1,
normalised so that µ(S1) = 1.
Theorem 6.5. Let f0 and f1 be two loxodromic elements in Aut(D) that are not antipar-
allel, and let X and Y be the sets described above, which depend on f0 and f1. If Y is
non-empty, then µ(E≥ℵ0) = µ(X).
6. CODING LIMIT SETS 151
Proof. By Lemma 6.1 we have
X \ E≥ℵ0 =∞⋃n=0
(X \
∞⋃k=n
i∈0,1k
Yi
).
We write
Qn = X \∞⋃k=n
i∈0,1k
Yi
and show that each Qn has zero measure. Suppose towards contradiction that some Qn
has positive measure, then by the Lebesgue density theorem 4.7 there exists x ∈ Qn such
that
(26) limr→0
µ(Qn ∩Bx(r))
µ(Bx(r))= 1.
Since x lies in X, x has at least one address and so we can choose some i ∈ 0, 1ω such
that x = x(i). For each positive integer m we write rm = |Xim|. Since x(i) ∈ Xim we
have Yim ⊆ Xim ⊂ Bx(rm) for all m ≥ n. Since Qn ⊆ X \Yim and µ(Bx(rm)) = 2|Xim|
we haveµ(Qn ∩Bx(rm))
µ(Bx(rm))≤
2|Xim| − |Yim|2|Xim|
for all m ≥ n. By Lemma 6.4 there is some constant c > 0 such that for all m we have
|Yim|/|Xim| ≥ c. Henceµ(Qn ∩Bx(rm))
µ(Bx(rm))≤ 1− c
2
for all m ≥ n. Upon taking the limit as m→∞, we obtain a contradiction with (26), and
so Qn must have zero measure after all.
Lemma 6.6. Let f0 and f1 be two loxodromic elements in Aut(D) that are not antiparallel,
then we have
E>ℵ0 = f0(E>ℵ0) ∪ f1(E>ℵ0).
Proof. First suppose z ∈ E>ℵ0 , that is z has uncountably many addresses. This
means that there is an uncountable set C ⊆ 0, 1ω such that for all i ∈ C we have
z = x((i0, . . . , in, . . .)). Hence for j = 0, 1 we have fj(z) = x((j, i0, . . . , in, . . .)) for all
i ∈ C. This means that fj(z) ∈ E>ℵ0 and so E>ℵ0 ⊇ f0(E>ℵ0) ∪ f1(E>ℵ0). Conversely,
152 6. CODING LIMIT SETS
again suppose z ∈ E>ℵ0 so that there is an uncountable set C ⊆ 0, 1ω, such that for
all i ∈ C we have z = x((i0, . . . , in, . . .)). By the pigeon-hole principle, either i0 = 0 or
i0 = 1 for uncountably many i ∈ C; hence either f−10 (z) ∈ E>ℵ0 or f−11 (z) ∈ E>ℵ0 . In
other words z ∈ f0(E>ℵ0) ∪ f1(E>ℵ0).
Theorem 6.7. Let f0 and f1 be two loxodromic elements in Aut(D) that are not antipar-
allel, and let X and Y be the sets described above, which depend on f0 and f1. If Y is
non-empty and E>ℵ0 is measurable, then either µ(E>ℵ0) = µ(X) or µ(Eℵ0) = µ(X).
Proof. By Lemma 6.1 we have X = Λ+, and since the forward and backward limit
sets are disjoint, Λ+ = Λ+c by Theorem 4.18. Furthermore, by Lemma 6.6, E>ℵ0 is
forward invariant. Notice that E≤ℵ0 is backward invariant since E>ℵ0 is forward invariant.
Hence it follows from Lemma 4.8 (applied to S−1 rather than S) that µ(Λ+c ∩ E≤ℵ0) > 0
implies µ(E≤ℵ0) = 1. That is if µ(X ∩ E≤ℵ0) > 0, then µ(X ∩ E≤ℵ0) = µ(X), and so
µ(Eℵ0) = µ(X) by Theorem 6.5. Now either µ(X ∩ Eℵ0) > 0 so that µ(Eℵ0) = µ(X),
otherwise µ(X ∩ Eℵ0) = 0, and so µ(Eℵ0) = 0 since Eℵ0 ⊆ X.
If S is not free then we shall show that E>ℵ0 is not empty. When Y = f0(X)∩ f1(X) has
empty interior the semigroup S = 〈f0, f1〉 is free. We show that for at least some choices
of f0 and f1, S is not free. We go on to show that whenever S is not free, µ(E2ℵ0 ) = µ(X).
We first observe that by conjugating, we can ensure that the axes of the generators are
symmetrical about both the real and imaginary axes, as shown in Figure 6.1. Hence there
are only three variables in R+ that determine S up to conjugation: the distance between
the axes of the generators and the two translation lengths of the generators. Let `h and
`v denote the geodesics contained in the horizontal and vertical axes respectively. Let σh
and σv denote the reflections in `h and `v respectively. If w is the element
w = (i0, . . . , in−1)
in 0, 1<ω then we let w∗ denote the element obtained from w by changing each instance
of 0 to 1 and each instance of 1 to 0. That is
w∗ = (1− i0, . . . , 1− in−1).
6. CODING LIMIT SETS 153
We let w† be the element obtained from w by reversing the order of each letter of w. That
is
w† = (in−1, . . . , i0).
We say that an element w in 0, 1<ω is palindromic if w = w†. For a loxodromic element
g, we let Ax(g) denote the axis of g with its direction marked. This is in contrast with
the notation ax(g), which denotes the axis of g as a set.
Lemma 6.8. If w is an element in 0, 1<ω, then σhFwσh = F−1w†
. Moreover, if f0 and f1
have the same translation lengths, then σvFwσv = Fw∗.
Proof. By considering the action of σhfiσh on the point of intersection of ax(fi) and
`h we see that σhfiσh = f−1i for i = 0, 1. Hence the first statement follows upon observing
that if g = fi1fi2 . . . fin−1fin then
σhgσh = (σhfi1σh)(σhfi2σh) . . . (σhfin−1σh)(σhfinσh).
The second statement is similar. Noting that both generators have the same translation
length and considering, for example, the action of σvfiσv on the intersection point of
ax(f1−i) and `h, we see that σvfiσv = f1−i.
Suppose now that the generators have equal translation lengths and g is an element of S
such that g = Fw for some palindromic element w in 0, 1<ω, and write g∗ = Fw∗ . For
example g could be f0f1f0, and so g∗ = f1f0f1. A consequence of the first statement in the
lemma is that Ax(g) = σh(Ax(g−1)). It follows that Ax(g) is orthogonal to `h. Similarly,
by the second statement in the lemma we see that Ax(g) = σv(Ax(g∗)). Figure 6.2 shows
an example in the case where the axes of f0 and f1 do not meet.
Since σvgσv = g∗ it follows that g and g∗ have the same translation lengths. This means
that if we can find g induced by a palindromic element w such that w 6= w∗ and whose axis
is equal to `h, then g ∈ S is induced by at least two different elements of 0, 1<ω, and so S
is not a free semigroup. Towards this, we consider varying the common translation length
τ of the generators, and consider g and g∗ as dependant on τ . Choose τ small enough
such that f0(b) is in the right half-plane. Now choose n large enough such that fn1 f0(0) is
154 6. CODING LIMIT SETS
`v
`h
f1f0
g g∗
Figure 6.2. The axis of an element g in S induced by a palindromic element.
close enough to f0(b) (where b is the attracting fixed point of f1) such that f0fn1 f0(0) is in
the right half-plane, and close to the ideal boundary. It follows that the attracting fixed
point of f0fn1 f0 is in the right half-plane. We put gn = f0f
n1 f0 and conclude that the axis
of gn is contained in the right half-plane. Now, with n fixed, we see that as τ tends to
∞ the attracting fixed point of gn converges to a, the attracting fixed point of f0. Hence
for large τ , the axis of g is in the left half-plane. Since the attracting and repelling fixed
points of g vary continuously with τ we see by the intermediate value theorem that for
some τ the axis of g coincides with `v. In this case we must have g = g∗ as g and g∗ have
the same translation lengths. Since w and w∗ are distinct and g = Fw = Fw∗ , we see that
S is not a free semigroup.
The above example of a semigroup that is not free was found by relying on symmetry; we
do not know if a ‘typical’ choice of f0 and f1 gives rise to a semigroup that is not free.
However, whenever a semigroup is not free, we shall see that the set of points in X with
2ℵ0-many addresses has the same measure as X.
For any p in 0, 1<ω, let A(p) be the set of points x ∈ X such that every address for x
eventually avoids p. That is, A(p) is the set of those x in X with the property that for all
i ∈ 0, 1ω such that x(i) = x, we have (in+1, . . . , in+|p|) = p for only finitely many n ∈ N.
Recall that for i and j in 0, 1<ω we let i, j denote the concatenation of i and j.
Lemma 6.9. For every p in 0, 1<ω, the set A(p) has measure zero.
6. CODING LIMIT SETS 155
Proof. The proof is very similar to that of Theorem 6.5. We let Am(p) denote the set
of points x ∈ X such that every address of x avoids p after m letters. That is, if x = x(i)
for some i ∈ 0, 1ω, then i m, that is (i0, . . . , im−1), does not contain p. Hence
Am(p) = X \⋃
i∈0,1m
j∈0,1<ω
Xi,j,p.
We now suppose towards contradiction that Am(p) has positive measure, so that by the
Lebesgue density theorem there exists a point x ∈ Am(p) such that
(27) limr→0
µ(Am(p) ∩Bx(r))
µ(Bx(r))= 1,
where as before, Bx(r) is the ball of radius r centred at x. Choose some address i of x.
For each positive integer n we write rn = |Xin|. Since x(i) ∈ Xin we have, for all n ≥ m,
Xin,p ⊆ Xin ⊂ Bx(rn). Hence we have
µ(Am(p) ∩Bx(rn))
µ(Bx(rn))≤
2|Xin| − |Xin,p|2|Xin|
for all n ≥ m. We claim there is a constant c > 0 such that for all n we have |Xin,p|/|Xin| ≥
c. Assume for now the claim is true, then
µ(Am(p) ∩Bx(rn))
µ(Bx(rn))≤ 1− c
2
for all n ≥ m. Upon taking the limit as n→∞, we obtain a contradiction with (27), and
so Am(p) must have zero measure after all. It follows that
A(p) =⋃m∈ω
Am(p)
also has zero measure.
In order to verify the constant c exists as claimed, we apply equation (25) from the proof
of Lemma 6.4 to give
|Xin,p||Xin|
1− |Fin,p(0)|2
1− |Fin(0)|2.
In the disc model, for any x ∈ D we have
ρ(0, x) = log
(1 + |x|1− |x|
)
156 6. CODING LIMIT SETS
(see for example [34, Section 1.6]), and so exp[−ρ(0, x)] 1− |x| as x runs along any se-
quence of points tending to the ideal boundary. Moreover, for any Mobius transformations
F and g we have ρ(Fg(0), 0) = ρ(g(0), F−1(0)) and
ρ(F−1(0), 0)− ρ(f(0), 0) ≤ ρ(g(0), F−1(0)) ≤ ρ(F−1(0), 0) + ρ(f(0), 0).
It follows that as n→∞,
1− |Fin(0)|2 exp[−ρ(Fin(0), 0)] exp[−ρ(Fin,p(0), 0)] 1− |Fin,p(0)|2.
It follows that |Xin,p|/|Xin| is bounded between two positive real numbers for all large
enough n, and so the constant c exists as claimed.
Lemma 6.10. Suppose S is not free, say for some distinct p and q in 0, 1<ω we have
Fp = Fq. Then, for any i ∈ 0, 1ω that does not eventually avoid p, the point x(i) in S
has 2ℵ0-many addresses; that is, x(i) ∈ E2ℵ0 .
Proof. Suppose i ∈ 0, 1ω takes the form i = j1, p, j2, p, . . . , jn, p, . . . where each
jk ∈ 0, 1<ω. This representation need not be unique, and possibly some jk have length
0. For any k ∈ 0, 1ω consider the element i(k) ∈ 0, 1ω obtained from i by replacing
the instance of p immediately after jn with q if k(n) = 1; otherwise we leave it as p. If the
lengths |p| and |q| of p and q (respectively) are equal, it is clear that the map k 7→ i(k)
is one-to-one. Next consider the case where these lengths are not equal, say |p| < |q|.
Suppose towards contradiction that k 7→ i(k) is not one-to-one, say k and k′ are distinct
elements in 0, 1ω and yet i(k) = i(k′). If m is the least positive integer such that k(m)
disagrees with k′(m), then we have
j1, . . . , jm, p, . . . = j1, . . . , jm, q, . . . .
In particular q |p| = p, but then as Fq = Fp we have I = fq|p| . . . fq|q|−1, which is
impossible. Hence in all cases, the map k 7→ i(k) is one-to-one. It follows that there are
2ℵ0-many addresses for x(i).
The discussion following Lemma 6.8 shows that there exist choices of f0 and f1 such that
S is not free. The next theorem tells us that, for such semigroups, almost every point in
X has 2ℵ0-many addresses.
6. CODING LIMIT SETS 157
Theorem 6.11. If S = 〈f0, f1〉 is not free, then E2ℵ0 has the same measure as X.
Proof. As S is not free, there are distinct p and q in 0, 1<ω such that Fp = Fq.
By Lemma 6.9 the set of points for which every address eventually avoids the block p has
measure zero. Hence X \A(p), the set of points in X for which there exists some address
that contains infinitely many instances of the block p, has the same measure as X. By
Lemma 6.10 X \ A(p) is contained in E2ℵ0 , and so E2ℵ0 also has the same measure as
X.
Although the theorem above can be generalised to higher dimensions, it is not clear that
it can be generalised to cases where Λ+(S) has non-integer Hausdorff dimension; for if s
is the non-integer Hausdorff dimension of Λ+, then the direct analogue of the Lebesgue
density theorem fails spectacularly when Λ+ has finite and nonzero s-measure.
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